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            Lecture 24: Large Deviations
         
    
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<p><strong>Description:</strong> Covers large deviation. Like expectation, it gives three other notions in solving bounds and many frequently experienced problems in computer science, such as determining the probability a random variable will deviate from its expectation.</p> <p><strong>Speaker:</strong> Tom Leighton</p>
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<p><span m="360">The</span> <span m="490">following</span> <span m="940">content</span> <span m="1530">is</span> <span m="1650">provided</span> <span m="2100">under</span> <span m="2350">a</span> <span m="2400">Creative</span> <span m="2800">Commons</span> <span m="3220">license.</span> <span m="4340">Your</span> <span m="4510">support</span> <span m="5050">will</span> <span m="5200">help</span> <span m="5420">MIT</span> <span m="6130">OpenCourseWare</span> <span m="6660">continue</span> <span m="7170">to</span> <span m="7290">offer</span> <span m="7690">high</span> <span m="7910">quality</span> <span m="8420">educational</span> <span m="9060">resources</span> <span m="9640">for</span> <span m="9820">free.</span> <span m="11020">To</span> <span m="11040">make</span> <span m="11250">a</span> <span m="11290">donation</span> <span m="11980">or</span> <span m="12230">view</span> <span m="12680">additional</span> <span m="13120">materials</span> <span m="13640">from</span> <span m="13830">hundreds</span> <span m="14240">of</span> <span m="14360">MIT</span> <span m="14760">courses,</span> <span m="15840">visit</span> <span m="16100">MIT</span> <span m="16510">OpenCourseWare</span> <span m="17560">at</span> <span m="17750">ocw.mit.edu.</span> </p>
<p><span m="23575">PROFESSOR: So</span> <span m="24120">today,</span> <span m="24420">we're</span> <span m="24540">going</span> <span m="24660">to</span> <span m="24800">talk</span> <span m="25390">about</span> <span m="25740">the</span> <span m="25830">probability</span> <span m="27160">that</span> <span m="27290">a</span> <span m="27380">random</span> <span m="27860">variable</span> <span m="28800">deviates</span> <span m="29610">by</span> <span m="29810">a</span> <span m="29870">certain</span> <span m="30250">amount</span> <span m="30770">from</span> <span m="31100">its</span> <span m="31240">expectation.</span> <span m="32850">Now,</span> <span m="33000">we've</span> <span m="33170">seen</span> <span m="33440">examples</span> <span m="34170">where</span> <span m="35120">a</span> <span m="35250">random</span> <span m="35610">variable</span> <span m="36050">is</span> <span m="36170">very</span> <span m="36430">unlikely</span> <span m="37180">to</span> <span m="37290">deviate</span> <span m="37900">much</span> <span m="38260">from</span> <span m="38390">its</span> <span m="38530">expectation.</span> <span m="39790">For</span> <span m="39920">example,</span> <span m="40400">if</span> <span m="40490">you</span> <span m="40580">flip</span> <span m="40900">100</span> <span m="41470">mutually</span> <span m="41910">independent</span> <span m="42560">fair</span> <span m="42840">coins,</span> <span m="44040">you're</span> <span m="44220">very</span> <span m="44700">likely</span> <span m="45060">to</span> <span m="45160">wind</span> <span m="45500">up</span> <span m="45630">with</span> <span m="45780">close</span> <span m="46250">to</span> <span m="46340">50</span> <span m="46740">heads,</span> <span m="47810">very</span> <span m="48130">unlikely</span> <span m="48660">to</span> <span m="48740">wind</span> <span m="49040">up</span> <span m="49220">with</span> <span m="49940">25</span> <span m="50550">or</span> <span m="50640">fewer</span> <span m="50990">heads,</span> <span m="51290">for</span> <span m="51430">example.</span> </p>
<p><span m="52820">We've</span> <span m="53000">also</span> <span m="53270">seen</span> <span m="53460">examples</span> <span m="54010">of</span> <span m="54100">distributions</span> <span m="54950">where</span> <span m="55780">you</span> <span m="55980">are</span> <span m="56230">very</span> <span m="56610">likely</span> <span m="57070">to</span> <span m="57230">be</span> <span m="57360">far</span> <span m="57740">from</span> <span m="58100">your</span> <span m="58510">expectation,</span> <span m="59780">for</span> <span m="59920">example,</span> <span m="60430">that</span> <span m="60610">problem</span> <span m="61100">when</span> <span m="61230">we</span> <span m="61360">had</span> <span m="61590">the</span> <span m="61810">communications</span> <span m="62540">channel,</span> <span m="63060">and</span> <span m="63160">we</span> <span m="63260">were</span> <span m="63360">measuring</span> <span m="63570">the</span> <span m="63650">latency</span> <span m="64310">of</span> <span m="64400">a</span> <span m="64459">packet</span> <span m="64890">crossing</span> <span m="65290">the</span> <span m="65360">channel.</span> <span m="66500">There,</span> <span m="67210">most</span> <span m="67570">of</span> <span m="67640">the</span> <span m="67710">time,</span> <span m="68670">your</span> <span m="68900">latency</span> <span m="69340">would</span> <span m="69430">be</span> <span m="69530">10</span> <span m="69800">milliseconds.</span> <span m="71360">But</span> <span m="71550">the</span> <span m="71650">expected</span> <span m="72250">latency</span> <span m="72720">was</span> <span m="72890">infinite.</span> <span m="73730">So</span> <span m="74320">you're</span> <span m="74510">very</span> <span m="74910">likely</span> <span m="75310">to</span> <span m="75400">deviate</span> <span m="76060">a</span> <span m="76180">lot</span> <span m="76640">from</span> <span m="76840">your</span> <span m="76970">expectation</span> <span m="77850">in</span> <span m="78000">that</span> <span m="78180">case.</span> </p>
<p><span m="79650">Last</span> <span m="80010">time,</span> <span m="80880">we</span> <span m="81260">looked</span> <span m="81440">at</span> <span m="81530">the</span> <span m="81590">variance.</span> <span m="82380">And</span> <span m="82500">we</span> <span m="82580">saw</span> <span m="82810">how</span> <span m="82960">that</span> <span m="83130">gave</span> <span m="83350">us</span> <span m="83460">some</span> <span m="83870">feel</span> <span m="84790">for</span> <span m="84920">the</span> <span m="85010">likelihood</span> <span m="86130">of</span> <span m="86460">being</span> <span m="86770">far</span> <span m="87540">from</span> <span m="87730">the</span> <span m="87830">expectation--</span> <span m="88650">high</span> <span m="88970">variance</span> <span m="89540">meaning</span> <span m="89900">you're</span> <span m="90450">more</span> <span m="90810">likely</span> <span m="91210">to</span> <span m="91310">deviate</span> <span m="92190">from</span> <span m="92400">the</span> <span m="92480">expectation.</span> <span m="93930">Today,</span> <span m="94360">we're</span> <span m="94490">going</span> <span m="94640">to</span> <span m="94710">develop</span> <span m="95480">specific</span> <span m="96100">tools</span> <span m="96640">for</span> <span m="96770">bounding</span> <span m="97810">or</span> <span m="97980">limiting</span> <span m="98500">the</span> <span m="98570">probability</span> <span m="99580">you</span> <span m="99750">deviate</span> <span m="100370">by</span> <span m="100630">a</span> <span m="100710">specified</span> <span m="101270">amount</span> <span m="101630">from</span> <span m="101760">the</span> <span m="101840">expectation.</span> <span m="103130">And</span> <span m="103270">the</span> <span m="103330">first</span> <span m="103620">tool</span> <span m="103860">is</span> <span m="103960">known</span> <span m="104240">as</span> <span m="104370">Markov's</span> <span m="104960">theorem.</span> </p>
<p><span m="106090">Markov's</span> <span m="106530">theorem</span> <span m="107030">says</span> <span m="107430">that</span> <span m="107560">if the</span> <span m="107680">random</span> <span m="108080">variable</span> <span m="108570">is</span> <span m="108790">always</span> <span m="109680">non-negative,</span> <span m="111460">then</span> <span m="112170">it</span> <span m="112310">is</span> <span m="112550">unlikely</span> <span m="113780">to</span> <span m="113910">greatly</span> <span m="114530">exceed</span> <span m="115200">its</span> <span m="115410">expectation.</span> <span m="116510">In</span> <span m="116670">particular,</span> <span m="127090">if</span> <span m="127380">R</span> <span m="129479">is</span> <span m="130009">a</span> <span m="130130">non-negative</span> <span m="136570">random</span> <span m="136970">variable,</span> <span m="140820">then</span> <span m="141100">for</span> <span m="141310">all</span> <span m="141600">x</span> <span m="142230">bigger</span> <span m="142470">than</span> <span m="142620">0,</span> <span m="147610">the</span> <span m="147730">probability</span> <span m="148770">that</span> <span m="148970">R</span> <span m="150190">is</span> <span m="150420">at</span> <span m="150560">least</span> <span m="150940">x</span> <span m="152640">is</span> <span m="152880">at</span> <span m="153010">most</span> <span m="154160">the</span> <span m="154280">expected</span> <span m="154940">value</span> <span m="155330">of</span> <span m="155450">R,</span> <span m="155705">the</span> <span m="155960">mean,</span> <span m="157210">divided</span> <span m="157720">by</span> <span m="157960">x.</span> </p>
<p><span m="159950">So</span> <span m="160220">in</span> <span m="160300">other</span> <span m="160440">words,</span> <span m="160620">if</span> <span m="160670">R</span> <span m="160950">is</span> <span m="161090">never</span> <span m="161960">negative--</span> <span m="164580">for</span> <span m="164700">example,</span> <span m="165200">say</span> <span m="165410">the</span> <span m="165510">expected</span> <span m="166080">value</span> <span m="166470">is</span> <span m="166560">smaller.</span> <span m="168290">Then</span> <span m="168470">the</span> <span m="168540">probability</span> <span m="169160">R</span> <span m="169500">is</span> <span m="169670">large</span> <span m="171280">will</span> <span m="171410">be</span> <span m="171520">a</span> <span m="171570">small</span> <span m="171900">number.</span> <span m="172240">Because</span> <span m="172450">I'll</span> <span m="172560">have a</span> <span m="172600">small</span> <span m="172950">number</span> <span m="173450">over</span> <span m="173730">a</span> <span m="173780">big</span> <span m="173970">number.</span> </p>
<p><span m="175700">So it</span> <span m="175810">says</span> <span m="176130">that</span> <span m="176290">you</span> <span m="176510">are</span> <span m="177170">unlikely</span> <span m="177830">to</span> <span m="177960">greatly</span> <span m="178460">exceed</span> <span m="179640">the</span> <span m="180030">expected</span> <span m="180490">value.</span> <span m="180900">So</span> <span m="181050">let's</span> <span m="181280">prove</span> <span m="181600">that.</span> <span m="185050">Now,</span> <span m="185320">from</span> <span m="185670">the</span> <span m="186740">theorem</span> <span m="187140">of</span> <span m="187240">total</span> <span m="187620">expectation</span> <span m="188610">that</span> <span m="188840">you</span> <span m="189000">did</span> <span m="189180">in</span> <span m="189270">recitation</span> <span m="189900">last</span> <span m="190190">week,</span> <span m="191230">we</span> <span m="191400">can</span> <span m="191550">compute</span> <span m="191980">the</span> <span m="192080">expected</span> <span m="192690">value</span> <span m="193040">of</span> <span m="193150">R</span> <span m="193690">by</span> <span m="193800">looking</span> <span m="194080">at</span> <span m="194170">two</span> <span m="194360">cases--</span> <span m="198360">the</span> <span m="198480">case</span> <span m="198940">when</span> <span m="199070">R</span> <span m="199380">is</span> <span m="199940">at</span> <span m="200110">least</span> <span m="200420">x,</span> <span m="208690">and</span> <span m="209010">the</span> <span m="209080">case</span> <span m="209520">when</span> <span m="209680">R</span> <span m="210070">is</span> <span m="210260">less</span> <span m="210510">than</span> <span m="210660">x.</span> <span m="221860">That's</span> <span m="222170">from</span> <span m="222380">the</span> <span m="222450">theorem</span> <span m="222780">of</span> <span m="222870">total</span> <span m="223180">expectation.</span> </p>
<p><span m="223810">I</span> <span m="223840">look</span> <span m="224000">at</span> <span m="224070">two</span> <span m="224220">cases.</span> <span m="225500">R</span> <span m="225760">is</span> <span m="225890">bigger</span> <span m="226180">than</span> <span m="226310">x.</span> <span m="226560">Take</span> <span m="226740">the</span> <span m="227120">expected</span> <span m="227640">value</span> <span m="227950">there</span> <span m="228200">times</span> <span m="228520">the</span> <span m="228590">probability</span> <span m="229250">of</span> <span m="229330">this</span> <span m="229550">case</span> <span m="229840">happening</span> <span m="230880">plus</span> <span m="231310">the</span> <span m="231410">case</span> <span m="231725">when</span> <span m="232040">R is</span> <span m="232480">less</span> <span m="232710">than</span> <span m="232830">x.</span> <span m="234900">OK,</span> <span m="235460">now</span> <span m="237130">since</span> <span m="237480">R</span> <span m="237750">is</span> <span m="237880">non-negative,</span> <span m="238870">this</span> <span m="239180">is</span> <span m="239270">at</span> <span m="239380">least</span> <span m="239780">0.</span> <span m="241950">R</span> <span m="242200">can't</span> <span m="242560">ever</span> <span m="242730">be</span> <span m="242880">negative.</span> <span m="243320">So</span> <span m="243440">the</span> <span m="243530">expectation</span> <span m="244140">can't</span> <span m="244370">be</span> <span m="244460">negative.</span> <span m="245380">A</span> <span m="245480">probability</span> <span m="246090">can't</span> <span m="246350">be</span> <span m="246450">negative.</span> <span m="246840">So</span> <span m="247340">this</span> <span m="247560">is</span> <span m="247640">at</span> <span m="247780">least</span> <span m="248150">0.</span> </p>
<p><span m="250210">And</span> <span m="251370">this</span> <span m="251750">is</span> <span m="251920">trivially</span> <span m="253370">at</span> <span m="253540">least</span> <span m="253830">x.</span> <span m="255500">Because</span> <span m="256410">I'm</span> <span m="256560">taking</span> <span m="256839">the</span> <span m="256920">expected</span> <span m="257420">value</span> <span m="257709">of</span> <span m="257800">R</span> <span m="258170">in</span> <span m="258300">the</span> <span m="258370">case</span> <span m="258760">when R is</span> <span m="259230">at</span> <span m="259329">least</span> <span m="259610">x.</span> <span m="260850">So</span> <span m="261769">R</span> <span m="262040">is</span> <span m="262160">always</span> <span m="262500">at</span> <span m="262590">least</span> <span m="262860">x</span> <span m="263470">in</span> <span m="263560">this</span> <span m="263710">case.</span> <span m="263980">So</span> <span m="264080">its</span> <span m="264190">expected</span> <span m="264620">value is</span> <span m="264990">at</span> <span m="265090">least</span> <span m="265360">x.</span> <span m="267680">So</span> <span m="267960">that</span> <span m="268160">means</span> <span m="268500">that</span> <span m="268980">the</span> <span m="269110">expected</span> <span m="269570">value</span> <span m="269860">of</span> <span m="269920">R</span> <span m="270220">is</span> <span m="270370">at</span> <span m="270510">least</span> <span m="271610">x</span> <span m="272790">times</span> <span m="273140">the</span> <span m="273210">probability</span> <span m="273780">R is</span> <span m="274100">greater</span> <span m="274380">than</span> <span m="274460">x,</span> <span m="276030">R is</span> <span m="276290">greater</span> <span m="276480">or equal</span> <span m="276760">to</span> <span m="276880">x.</span> </p>
<p><span m="279700">And</span> <span m="279860">now</span> <span m="280380">I</span> <span m="280480">can</span> <span m="280680">get</span> <span m="280850">the</span> <span m="280920">theorem</span> <span m="281380">by</span> <span m="281520">just</span> <span m="282910">dividing</span> <span m="283580">by</span> <span m="283820">x.</span> <span m="290230">I'm</span> <span m="290530">less</span> <span m="290850">than</span> <span m="291040">or</span> <span m="291150">equal</span> <span m="291430">to</span> <span m="292350">the</span> <span m="292440">expected</span> <span m="292970">value</span> <span m="293250">of</span> <span m="293360">R</span> <span m="295050">divided</span> <span m="295400">by</span> <span m="295550">x.</span> <span m="297980">So</span> <span m="298110">it's</span> <span m="298220">a</span> <span m="298260">very</span> <span m="298810">easy</span> <span m="299190">theorem</span> <span m="299540">to</span> <span m="299620">prove.</span> <span m="300060">But</span> <span m="300190">it's</span> <span m="300350">going</span> <span m="300470">to</span> <span m="300530">have</span> <span m="300810">amazing</span> <span m="301330">consequences</span> <span m="302080">that</span> <span m="302210">we're</span> <span m="302300">going</span> <span m="302420">to</span> <span m="302810">build</span> <span m="303200">up</span> <span m="303440">through</span> <span m="303610">a</span> <span m="303660">series</span> <span m="304120">of</span> <span m="305080">results</span> <span m="305480">today.</span> <span m="306850">Any</span> <span m="306930">questions</span> <span m="307550">about</span> <span m="308680">Markov's</span> <span m="309130">theorem</span> <span m="309710">and</span> <span m="309850">the</span> <span m="309910">proof?</span> </p>
<p><span m="314090">All</span> <span m="314260">right,</span> <span m="314520">there's</span> <span m="314720">a</span> <span m="314780">simple</span> <span m="315170">corollary,</span> <span m="316880">which</span> <span m="317120">is</span> <span m="317230">useful.</span> <span m="320710">Again,</span> <span m="321090">if</span> <span m="322000">R</span> <span m="322720">is</span> <span m="323400">a</span> <span m="323540">non-negative</span> <span m="326670">random</span> <span m="327100">variable,</span> <span m="329930">then</span> <span m="331120">for</span> <span m="331340">all</span> <span m="331740">c</span> <span m="332510">bigger</span> <span m="332820">than</span> <span m="333000">0,</span> <span m="334740">the</span> <span m="334860">probability</span> <span m="336110">that</span> <span m="336430">R</span> <span m="337150">is</span> <span m="337430">at</span> <span m="337520">least</span> <span m="337980">c</span> <span m="338470">times</span> <span m="339050">its</span> <span m="339260">expected</span> <span m="339770">value</span> <span m="342420">is</span> <span m="342640">at</span> <span m="342750">most</span> <span m="343230">1</span> <span m="343410">and</span> <span m="343560">c.</span> <span m="345440">So</span> <span m="345610">the</span> <span m="345690">probability</span> <span m="346400">you're</span> <span m="346540">twice</span> <span m="347080">your</span> <span m="347200">expected</span> <span m="347670">value</span> <span m="348100">is</span> <span m="348210">at</span> <span m="348340">most</span> <span m="348820">1/2.</span> </p>
<p><span m="350190">And</span> <span m="350610">the</span> <span m="350750">proof</span> <span m="351120">is</span> <span m="351260">very</span> <span m="351540">easy.</span> <span m="353670">We</span> <span m="353830">just</span> <span m="354320">set</span> <span m="355220">x</span> <span m="355610">to</span> <span m="355700">be</span> <span m="355840">equal</span> <span m="356140">to</span> <span m="356240">c</span> <span m="356630">times</span> <span m="356990">the</span> <span m="357080">expected</span> <span m="357540">value</span> <span m="357810">of</span> <span m="357930">R</span> <span m="360020">in</span> <span m="360200">the</span> <span m="360260">theorem.</span> <span m="364020">So</span> <span m="364240">I</span> <span m="364320">just</span> <span m="364630">plug</span> <span m="365035">in</span> <span m="366290">x</span> <span m="368070">is</span> <span m="368330">c</span> <span m="368700">times</span> <span m="369230">the</span> <span m="369540">expected</span> <span m="370000">value</span> <span m="370380">of</span> <span m="370500">R.</span> <span m="372570">And</span> <span m="372710">I</span> <span m="372800">get</span> <span m="372990">expected</span> <span m="373450">value</span> <span m="373700">of</span> <span m="373830">R</span> <span m="374090">over</span> <span m="374340">c</span> <span m="374640">times</span> <span m="375010">the</span> <span m="375070">expected</span> <span m="375470">value</span> <span m="375720">of</span> <span m="375830">R,</span> <span m="376240">which</span> <span m="376420">is</span> <span m="376510">1/c.</span> <span m="378330">So</span> <span m="378450">you</span> <span m="378510">just</span> <span m="378960">plug in</span> <span m="379350">that</span> <span m="379520">value</span> <span m="380250">in</span> <span m="380360">Markov's</span> <span m="380810">theorem,</span> <span m="381260">and</span> <span m="381430">it</span> <span m="382180">comes</span> <span m="382460">out.</span> </p>
<p><span m="384800">All</span> <span m="384920">right,</span> <span m="385040">let's</span> <span m="385250">do</span> <span m="385500">some</span> <span m="385710">examples.</span> <span m="391110">Let's</span> <span m="391260">let</span> <span m="391460">R</span> <span m="391810">be</span> <span m="392060">the</span> <span m="392640">weight</span> <span m="396430">of</span> <span m="396700">a</span> <span m="396920">random</span> <span m="397440">person</span> <span m="398870">uniformly</span> <span m="399530">selected.</span> <span m="404030">And</span> <span m="405850">I</span> <span m="406000">don't</span> <span m="406220">know</span> <span m="406490">what</span> <span m="406820">the</span> <span m="406890">distribution</span> <span m="407490">of</span> <span m="407570">weights</span> <span m="407940">is</span> <span m="408240">in</span> <span m="408370">the</span> <span m="408440">country.</span> <span m="409930">But</span> <span m="410120">suppose</span> <span m="410800">that</span> <span m="411090">the</span> <span m="411190">expected</span> <span m="411780">value</span> <span m="412170">of</span> <span m="412290">R,</span> <span m="412690">which</span> <span m="412900">is</span> <span m="413600">the</span> <span m="413790">average</span> <span m="414250">weight,</span> <span m="415150">is</span> <span m="415350">100</span> <span m="415670">pounds.</span> <span m="417790">So</span> <span m="419140">if</span> <span m="419230">I</span> <span m="419390">average</span> <span m="419820">over</span> <span m="420030">all</span> <span m="420170">people,</span> <span m="420550">their</span> <span m="420710">weight</span> <span m="420940">is</span> <span m="421030">100</span> <span m="421360">pounds.</span> </p>
<p><span m="423810">And</span> <span m="423970">suppose</span> <span m="424410">I</span> <span m="424500">want</span> <span m="424730">to</span> <span m="424790">know</span> <span m="425900">the</span> <span m="425990">probability</span> <span m="427250">that</span> <span m="427400">the</span> <span m="427500">random</span> <span m="427830">person</span> <span m="428550">weighs</span> <span m="428890">at</span> <span m="429000">least</span> <span m="429250">200</span> <span m="429670">pounds.</span> <span m="433270">What</span> <span m="433430">can</span> <span m="433560">I</span> <span m="433620">say</span> <span m="433970">about</span> <span m="434250">that</span> <span m="434410">probability?</span> <span m="440820">Do</span> <span m="440970">I</span> <span m="441050">know</span> <span m="441210">it</span> <span m="441320">exactly?</span> <span m="445960">I</span> <span m="446030">don't</span> <span m="446210">think</span> <span m="446400">so.</span> <span m="446600">Because</span> <span m="446750">I</span> <span m="446840">don't</span> <span m="447050">know</span> <span m="447310">what</span> <span m="447450">the</span> <span m="447520">distribution</span> <span m="448080">of</span> <span m="448150">weights</span> <span m="448490">is.</span> </p>
<p><span m="449280">But</span> <span m="449470">I</span> <span m="449540">can</span> <span m="449690">still</span> <span m="450000">get</span> <span m="450180">an</span> <span m="450280">upper</span> <span m="450470">bound</span> <span m="450830">on</span> <span m="450940">this</span> <span m="451080">probability.</span> <span m="453530">What</span> <span m="453710">bound</span> <span m="453970">can</span> <span m="454090">I</span> <span m="454180">get</span> <span m="454400">on</span> <span m="454480">the</span> <span m="454560">probability</span> <span m="455180">that</span> <span m="455500">a</span> <span m="455640">random</span> <span m="455920">person</span> <span m="456210">has</span> <span m="456340">a</span> <span m="456400">weight</span> <span m="456580">of</span> <span m="456660">200</span> <span m="457150">given</span> <span m="457410">the</span> <span m="457480">facts</span> <span m="457850">here?</span> <span m="458090">Yeah.</span> </p>
<p><span m="459302">AUDIENCE:</span> <span m="459783">[INAUDIBLE]</span> </p>
<p></p>
<p><span m="466050">PROFESSOR: Yes,</span> <span m="467190">well,</span> <span m="467370">it's</span> <span m="467680">100</span> <span m="468080">over</span> <span m="468220">200,</span> <span m="468570">right.</span> <span m="468760">It's at</span> <span m="468940">most</span> <span m="469860">the</span> <span m="469970">expected</span> <span m="470510">value,</span> <span m="470890">which</span> <span m="470940">is</span> <span m="471030">100,</span> <span m="471880">over</span> <span m="472160">the</span> <span m="472320">x,</span> <span m="472590">which</span> <span m="472770">is</span> <span m="472880">200.</span> <span m="474420">And</span> <span m="474570">that's</span> <span m="474870">equal</span> <span m="475170">to</span> <span m="475350">1/2.</span> <span m="477390">So</span> <span m="477590">the</span> <span m="477680">probability</span> <span m="478330">that</span> <span m="478440">a</span> <span m="478470">random</span> <span m="478730">person</span> <span m="479090">weighs</span> <span m="479310">200</span> <span m="479690">pounds</span> <span m="480040">or</span> <span m="480090">more</span> <span m="480440">is</span> <span m="480590">at</span> <span m="480720">most</span> <span m="481330">1/2.</span> </p>
<p><span m="482230">Or</span> <span m="482480">I</span> <span m="482550">could</span> <span m="482690">plug</span> <span m="482950">it</span> <span m="483030">in</span> <span m="483580">here.</span> <span m="484940">The</span> <span m="485080">expected</span> <span m="485560">value</span> <span m="485850">is</span> <span m="485940">100.</span> <span m="487796">200</span> <span m="488260">is</span> <span m="488370">twice</span> <span m="488770">that.</span> <span m="489020">So</span> <span m="489150">c</span> <span m="489430">would</span> <span m="489570">be</span> <span m="489700">2</span> <span m="490810">here.</span> <span m="491710">So</span> <span m="491780">the</span> <span m="491860">probability</span> <span m="492350">of</span> <span m="492420">being</span> <span m="492600">twice</span> <span m="492990">the</span> <span m="493070">expectation</span> <span m="493740">is</span> <span m="493840">at</span> <span m="493950">most</span> <span m="494360">1/2.</span> </p>
<p><span m="495190">Now</span> <span m="495270">of</span> <span m="495330">course,</span> <span m="495570">I'm</span> <span m="495710">using</span> <span m="496050">the</span> <span m="496120">fact</span> <span m="497220">that</span> <span m="497470">weight</span> <span m="497910">is</span> <span m="498120">never</span> <span m="498410">negative.</span> <span m="499280">That's</span> <span m="500110">obviously</span> <span m="500530">true.</span> <span m="500990">But</span> <span m="501010">it</span> <span m="501420">is</span> <span m="501510">implicitly</span> <span m="502600">being</span> <span m="502820">used</span> <span m="503110">here.</span> <span m="505400">So</span> <span m="507070">what</span> <span m="507430">fraction</span> <span m="508050">of</span> <span m="508110">the</span> <span m="508180">population</span> <span m="509000">now</span> <span m="510840">can</span> <span m="511100">weigh</span> <span m="511380">at</span> <span m="511480">least</span> <span m="511710">200</span> <span m="512010">pounds?</span> <span m="512909">Slightly</span> <span m="513370">different</span> <span m="513630">question.</span> <span m="514280">Before</span> <span m="514610">I</span> <span m="514740">asked</span> <span m="514929">you,</span> <span m="514990">if</span> <span m="515045">I</span> <span m="515100">take</span> <span m="515260">a</span> <span m="515340">random</span> <span m="515710">person,</span> <span m="517730">what's</span> <span m="517980">the</span> <span m="518059">probability</span> <span m="518539">they</span> <span m="518669">weigh</span> <span m="518870">at</span> <span m="518950">least</span> <span m="519150">200</span> <span m="519419">pounds?</span> <span m="520429">Now</span> <span m="520620">I'm</span> <span m="520820">asking,</span> <span m="522490">what</span> <span m="522770">fraction</span> <span m="523299">of</span> <span m="523360">the</span> <span m="523419">population</span> <span m="524720">can</span> <span m="524970">weigh</span> <span m="525290">at</span> <span m="525390">least</span> <span m="525600">200</span> <span m="525880">pounds</span> <span m="529950">if</span> <span m="530130">the average</span> <span m="530270">is</span> <span m="530800">100?</span> <span m="535280">What</span> <span m="535450">is</span> <span m="535590">it?</span> <span m="536594">Yeah?</span> </p>
<p><span m="537950">AUDIENCE: At</span> <span m="538100">most</span> <span m="538350">1/2.</span> </p>
<p><span m="538900">PROFESSOR: At</span> <span m="539070">most</span> <span m="539440">1/2.</span> <span m="539910">In</span> <span m="539990">fact,</span> <span m="540330">it's</span> <span m="540380">the</span> <span m="540460">same</span> <span m="541520">answer.</span> <span m="543300">And</span> <span m="545950">why?</span> <span m="548450">Why</span> <span m="548660">can't</span> <span m="548980">everybody</span> <span m="549420">weigh</span> <span m="549610">200</span> <span m="550000">pounds,</span> <span m="551460">so</span> <span m="551660">it</span> <span m="551720">would</span> <span m="551830">be</span> <span m="552000">all</span> <span m="552310">the</span> <span m="552420">population</span> <span m="553070">weighs</span> <span m="553290">200</span> <span m="553540">pounds</span> <span m="553980">at</span> <span m="554110">least?</span> </p>
<p><span m="555460">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="556990">PROFESSOR: Probability</span> <span m="557270">would</span> <span m="557550">be</span> <span m="557700">1,</span> <span m="558800">and</span> <span m="558950">that</span> <span m="559110">can't</span> <span m="559360">happen.</span> <span m="560690">And</span> <span m="560860">in</span> <span m="560980">fact,</span> <span m="561610">intuitively,</span> <span m="562820">if</span> <span m="562960">everybody</span> <span m="563350">weighs</span> <span m="563620">at</span> <span m="563700">least</span> <span m="563890">200</span> <span m="564190">pounds,</span> <span m="565040">the</span> <span m="565250">average</span> <span m="566240">is</span> <span m="566390">going</span> <span m="566520">to</span> <span m="566580">be</span> <span m="566670">at</span> <span m="566750">least</span> <span m="566970">200</span> <span m="567250">pounds.</span> <span m="567560">And we</span> <span m="567690">said</span> <span m="567870">the</span> <span m="567960">average</span> <span m="568320">was</span> <span m="568460">100.</span> <span m="570530">And</span> <span m="570630">this</span> <span m="570780">is</span> <span m="570910">illustrating</span> <span m="572240">this</span> <span m="572580">interesting</span> <span m="573050">thing</span> <span m="573370">that</span> <span m="574190">probability</span> <span m="575520">implies</span> <span m="576250">things</span> <span m="577680">about</span> <span m="578140">averages</span> <span m="578910">and</span> <span m="579100">fractions.</span> <span m="580000">Because</span> <span m="580290">it's</span> <span m="580450">really</span> <span m="580780">the</span> <span m="580870">same</span> <span m="581180">thing</span> <span m="581350">in</span> <span m="581440">disguise.</span> <span m="581960">The</span> <span m="582050">connection</span> <span m="582560">is,</span> <span m="582850">if</span> <span m="583140">I've</span> <span m="583300">got</span> <span m="583460">a</span> <span m="583490">bunch</span> <span m="583750">of</span> <span m="583960">people,</span> <span m="584500">say,</span> <span m="584690">in</span> <span m="584780">the</span> <span m="584860">country,</span> <span m="586310">I</span> <span m="586420">can</span> <span m="586560">convert</span> <span m="586880">a</span> <span m="586920">fraction</span> <span m="587760">that</span> <span m="587920">have</span> <span m="588100">some</span> <span m="588270">property</span> <span m="588720">into</span> <span m="588900">a</span> <span m="588940">probability</span> <span m="590190">by</span> <span m="590340">just</span> <span m="590560">selecting</span> <span m="591020">a</span> <span m="591090">random</span> <span m="591420">person.</span> <span m="592150">Yeah.</span> </p>
<p><span m="593150">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="601584">PROFESSOR: No,</span> <span m="602388">the</span> <span m="602790">variance</span> <span m="603580">could</span> <span m="603770">be</span> <span m="603910">very</span> <span m="604550">big.</span> <span m="605990">Because</span> <span m="606320">I</span> <span m="606390">might</span> <span m="606650">have</span> <span m="607010">a</span> <span m="607120">person</span> <span m="607510">that</span> <span m="607630">weighs</span> <span m="608310">a</span> <span m="608400">million</span> <span m="608960">pounds,</span> <span m="609890">say.</span> <span m="611150">So</span> <span m="611650">you</span> <span m="611730">have</span> <span m="611880">to</span> <span m="612600">get</span> <span m="612780">into</span> <span m="612950">that.</span> <span m="613110">But</span> <span m="613200">it</span> <span m="613350">gets</span> <span m="613750">a</span> <span m="613780">little</span> <span m="613960">bit</span> <span m="614080">more</span> <span m="614240">complicated.</span> <span m="615120">Yeah.</span> </p>
<p><span m="616390">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="619750">PROFESSOR: No,</span> <span m="620290">there's</span> <span m="620640">nothing</span> <span m="621060">being</span> <span m="621310">assumed</span> <span m="621660">about</span> <span m="621850">the</span> <span m="621920">distribution,</span> <span m="623370">nothing</span> <span m="623970">at</span> <span m="624060">all,</span> <span m="625880">OK?</span> <span m="626360">So</span> <span m="626500">that's</span> <span m="626740">the</span> <span m="626830">beauty</span> <span m="627200">of</span> <span m="627310">Markov's</span> <span m="627800">theorem.</span> <span m="628670">Well,</span> <span m="628960">I've</span> <span m="629330">assumed</span> <span m="629670">one</span> <span m="629930">thing.</span> <span m="630650">I</span> <span m="630790">assume</span> <span m="631330">that</span> <span m="631460">there</span> <span m="631550">is</span> <span m="631660">no</span> <span m="631870">negative</span> <span m="632360">values.</span> <span m="633254">That's</span> <span m="633701">it.</span> </p>
<p><span m="635489">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="639050">PROFESSOR: That's</span> <span m="639320">correct.</span> <span m="640120">They</span> <span m="640270">can</span> <span m="640390">distribute</span> <span m="640430">it</span> <span m="640730">any</span> <span m="640990">way</span> <span m="641120">with</span> <span m="641260">positive</span> <span m="641660">values.</span> <span m="642500">But</span> <span m="642820">we</span> <span m="643000">have</span> <span m="643230">a</span> <span m="643290">fact</span> <span m="643680">here</span> <span m="643780">we've</span> <span m="643960">used,</span> <span m="645150">that</span> <span m="645310">the</span> <span m="645470">average</span> <span m="645870">was</span> <span m="646000">100.</span> <span m="647490">So</span> <span m="647650">that</span> <span m="647890">does</span> <span m="648130">limit</span> <span m="648510">your</span> <span m="648620">distribution.</span> <span m="650620">In</span> <span m="650670">other</span> <span m="650820">words,</span> <span m="651090">you</span> <span m="651190">couldn't</span> <span m="651370">have</span> <span m="651530">a</span> <span m="651570">distribution</span> <span m="652160">where</span> <span m="652260">everybody</span> <span m="652650">weighs</span> <span m="652910">200</span> <span m="653230">pounds.</span> <span m="654520">Because</span> <span m="654720">then</span> <span m="654820">the</span> <span m="654950">average</span> <span m="655260">would</span> <span m="655350">be</span> <span m="655430">200,</span> <span m="655930">not</span> <span m="656230">100.</span> <span m="658380">But</span> <span m="658580">anything</span> <span m="659000">else</span> <span m="659550">where</span> <span m="660080">they're</span> <span m="660350">all</span> <span m="660520">positive</span> <span m="661060">and</span> <span m="661120">they</span> <span m="661220">average</span> <span m="661550">100,</span> <span m="662560">you</span> <span m="662710">know</span> <span m="663190">that</span> <span m="663310">at</span> <span m="663400">most</span> <span m="664150">half</span> <span m="664610">can</span> <span m="664750">be</span> <span m="664850">200.</span> <span m="666080">Because</span> <span m="666620">if</span> <span m="666720">you</span> <span m="666800">pick</span> <span m="666980">a</span> <span m="667060">random</span> <span m="667450">one,</span> <span m="667620">the</span> <span m="667690">probability</span> <span m="668270">of</span> <span m="668370">getting</span> <span m="669040">one</span> <span m="669270">that's</span> <span m="669430">200</span> <span m="669810">is</span> <span m="669900">at</span> <span m="670030">most</span> <span m="670620">1/2,</span> <span m="672200">which</span> <span m="672430">follows</span> <span m="672920">from</span> <span m="673250">Markov's</span> <span m="673740">theorem.</span> </p>
<p><span m="673860">And</span> <span m="673950">that's</span> <span m="674120">partly</span> <span m="674480">why</span> <span m="674690">it's</span> <span m="675790">so</span> <span m="676010">powerful.</span> <span m="677330">You</span> <span m="677460">didn't</span> <span m="677690">know</span> <span m="677990">anything</span> <span m="678180">about</span> <span m="678340">the</span> <span m="678410">distribution,</span> <span m="678980">really,</span> <span m="679340">except</span> <span m="680270">its</span> <span m="680470">expectation</span> <span m="681700">and</span> <span m="681960">that</span> <span m="682050">it</span> <span m="682110">was</span> <span m="682220">non-negative.</span> <span m="684470">Any</span> <span m="684650">other</span> <span m="684790">questions</span> <span m="686850">about</span> <span m="687100">this?</span> <span m="687975">I'll give you</span> <span m="688420">some</span> <span m="688690">more</span> <span m="688840">examples.</span> <span m="689690">All right,</span> <span m="690090">here's</span> <span m="690240">another</span> <span m="690450">example.</span> <span m="691850">Is</span> <span m="692070">it</span> <span m="692210">possible</span> <span m="692740">on</span> <span m="692860">the</span> <span m="692920">final</span> <span m="693270">exam</span> <span m="694700">for</span> <span m="694970">everybody</span> <span m="695480">in</span> <span m="695560">the</span> <span m="695630">class</span> <span m="696070">to</span> <span m="696190">do</span> <span m="696320">better</span> <span m="696640">than</span> <span m="696790">the</span> <span m="696860">mean</span> <span m="697220">score?</span> <span m="699670">No,</span> <span m="699950">of</span> <span m="700020">course</span> <span m="700300">not.</span> <span m="700550">Because</span> <span m="700850">if</span> <span m="700940">they</span> <span m="701070">did,</span> <span m="701320">the</span> <span m="701380">mean</span> <span m="701620">would</span> <span m="701740">be</span> <span m="701840">higher.</span> <span m="702760">Because</span> <span m="702940">the</span> <span m="703020">mean</span> <span m="703270">is</span> <span m="703380">the</span> <span m="703510">average.</span> </p>
<p><span m="704570">OK,</span> <span m="707125">let's do</span> <span m="707750">another</span> <span m="707990">example.</span> <span m="708430">Remember</span> <span m="708620">the</span> <span m="708950">Chinese</span> <span m="709440">appetizer</span> <span m="710020">problem?</span> <span m="711560">You're</span> <span m="711730">at</span> <span m="711840">the</span> <span m="711940">restaurant,</span> <span m="712420">big</span> <span m="712590">circular</span> <span m="713000">table.</span> <span m="713380">There's</span> <span m="713440">n</span> <span m="713670">people</span> <span m="713950">at the</span> <span m="714040">table.</span> <span m="715010">Everybody</span> <span m="715330">has</span> <span m="715460">one</span> <span m="715690">appetizer</span> <span m="716240">in</span> <span m="716320">front</span> <span m="716580">of</span> <span m="716640">them.</span> <span m="717250">And</span> <span m="717420">then</span> <span m="717670">the</span> <span m="717730">joker</span> <span m="718100">spins</span> <span m="718790">the</span> <span m="718890">thing</span> <span m="719110">in</span> <span m="719190">the</span> <span m="719250">middle</span> <span m="719440">of</span> <span m="719500">the</span> <span m="719560">table.</span> <span m="720520">So</span> <span m="720710">it</span> <span m="720780">goes</span> <span m="721060">around</span> <span m="721380">and</span> <span m="721470">around.</span> <span m="721850">And</span> <span m="721940">it</span> <span m="722020">stops</span> <span m="722520">in</span> <span m="722590">a</span> <span m="722680">random</span> <span m="723420">uniform</span> <span m="723870">position.</span> </p>
<p><span m="725140">And</span> <span m="725330">we</span> <span m="725450">wanted</span> <span m="725690">to</span> <span m="725880">know,</span> <span m="726180">what's</span> <span m="726420">the</span> <span m="726530">expected</span> <span m="727120">number</span> <span m="727460">of</span> <span m="727550">people</span> <span m="727860">to</span> <span m="727930">get</span> <span m="728070">the</span> <span m="728160">right</span> <span m="728370">appetizer</span> <span m="728900">back?</span> <span m="730530">What</span> <span m="730660">was</span> <span m="730780">the</span> <span m="730940">answer?</span> <span m="731280">Does</span> <span m="731380">anybody</span> <span m="731450">remember?</span> <span m="733460">One.</span> <span m="733825">So</span> <span m="734190">you</span> <span m="734300">expect</span> <span m="734660">one</span> <span m="734950">person</span> <span m="735480">to</span> <span m="735570">get</span> <span m="735720">the</span> <span m="735800">right</span> <span m="736000">appetizer</span> <span m="736530">back.</span> </p>
<p><span m="738290">Well,</span> <span m="738410">say</span> <span m="738610">I</span> <span m="738720">want</span> <span m="738950">to</span> <span m="739010">know</span> <span m="739110">the</span> <span m="739210">probability</span> <span m="739890">that</span> <span m="740400">all</span> <span m="740860">n</span> <span m="741180">people</span> <span m="741590">got</span> <span m="741790">the</span> <span m="741870">right</span> <span m="742070">appetizer</span> <span m="742620">back.</span> <span m="744150">What</span> <span m="744350">does</span> <span m="744530">Markov</span> <span m="745070">tell</span> <span m="745340">you</span> <span m="746100">about</span> <span m="746400">the</span> <span m="746480">probability</span> <span m="747300">that</span> <span m="747450">all</span> <span m="747820">n</span> <span m="748080">people</span> <span m="748460">get</span> <span m="748630">the</span> <span m="748720">right</span> <span m="748930">appetizer</span> <span m="749440">back?</span> <span m="751530">1/n.</span> <span m="752370">The</span> <span m="752470">expected</span> <span m="753000">value</span> <span m="753280">is</span> <span m="753390">1.</span> <span m="754680">And</span> <span m="755020">now</span> <span m="755210">you're</span> <span m="755360">asking</span> <span m="755670">the</span> <span m="755740">probability</span> <span m="756380">that</span> <span m="756550">you</span> <span m="756660">get</span> <span m="758700">R</span> <span m="759020">is at</span> <span m="759120">least</span> <span m="759450">n.</span> <span m="759705">So</span> <span m="759960">x</span> <span m="760180">is</span> <span m="760310">n.</span> <span m="760950">So</span> <span m="761130">it's</span> <span m="761230">1</span> <span m="761550">in</span> <span m="761610">n.</span> </p>
<p><span m="763230">And</span> <span m="763420">what</span> <span m="763610">was</span> <span m="763960">the</span> <span m="764050">probability,</span> <span m="764310">or</span> <span m="764610">what</span> <span m="764860">is</span> <span m="765170">the</span> <span m="765310">actual</span> <span m="765740">probability?</span> <span m="766270">In</span> <span m="766330">this</span> <span m="766480">case,</span> <span m="766700">you</span> <span m="766790">know</span> <span m="766910">the</span> <span m="766990">distribution,</span> <span m="767580">that</span> <span m="767730">everybody</span> <span m="768170">gets</span> <span m="768420">the</span> <span m="768490">right</span> <span m="768670">appetizer</span> <span m="769180">back,</span> <span m="769460">all n.</span> <span m="771300">1</span> <span m="771650">in</span> <span m="772010">n.</span> <span m="772690">So</span> <span m="772910">in</span> <span m="772990">the</span> <span m="773060">case</span> <span m="773450">of</span> <span m="774250">the</span> <span m="774360">Chinese</span> <span m="774750">appetizer</span> <span m="775330">problem,</span> <span m="776840">Markov's</span> <span m="777520">bound</span> <span m="777990">is</span> <span m="778110">actually</span> <span m="778520">the</span> <span m="778610">right</span> <span m="778860">answer,</span> <span m="779880">right</span> <span m="780130">on</span> <span m="780260">target,</span> <span m="781310">which</span> <span m="781780">gives</span> <span m="782400">you</span> <span m="783140">an</span> <span m="783260">example</span> <span m="783700">where</span> <span m="783820">you</span> <span m="783900">can't</span> <span m="784230">improve</span> <span m="784590">it.</span> </p>
<p><span m="785550">By</span> <span m="785770">itself,</span> <span m="786380">if</span> <span m="786550">you</span> <span m="786620">just</span> <span m="786920">know</span> <span m="787040">the</span> <span m="787120">expected</span> <span m="787550">value,</span> <span m="788730">there's</span> <span m="788920">no</span> <span m="789050">stronger</span> <span m="789600">theorem</span> <span m="790190">that</span> <span m="790510">way.</span> <span m="791290">Because</span> <span m="791570">Chinese</span> <span m="791950">appetizer</span> <span m="792750">is</span> <span m="792930">an</span> <span m="793010">example</span> <span m="793500">where</span> <span m="793670">the</span> <span m="793750">bound</span> <span m="794140">you</span> <span m="794230">get,</span> <span m="794810">1/n,</span> <span m="795970">of</span> <span m="796150">n</span> <span m="796280">people</span> <span m="796640">getting</span> <span m="796720">the</span> <span m="796780">right</span> <span m="796970">appetizer</span> <span m="797560">is</span> <span m="797800">in</span> <span m="797890">fact</span> <span m="798160">the</span> <span m="798220">true</span> <span m="798380">probability.</span> </p>
<p><span m="801530">OK,</span> <span m="802510">what</span> <span m="802760">about</span> <span m="803030">the</span> <span m="803120">hat</span> <span m="803440">check</span> <span m="803730">problem?</span> <span m="804630">Remember</span> <span m="805290">that?</span> <span m="805620">So there's</span> <span m="805850">n</span> <span m="806100">men</span> <span m="807070">put</span> <span m="807310">the</span> <span m="807390">hats</span> <span m="807730">in</span> <span m="807800">the</span> <span m="807870">coat</span> <span m="808110">closet.</span> <span m="809420">They</span> <span m="809590">get</span> <span m="809780">uniformly</span> <span m="810730">randomly</span> <span m="811210">scrambled.</span> <span m="811830">So</span> <span m="811960">it's</span> <span m="812090">a</span> <span m="812160">random</span> <span m="812630">permutation</span> <span m="813390">applied</span> <span m="813780">to</span> <span m="813840">the</span> <span m="813950">hats.</span> <span m="815530">Now</span> <span m="815780">each</span> <span m="815990">man</span> <span m="816160">gets</span> <span m="816340">a</span> <span m="816390">hat</span> <span m="816640">back.</span> <span m="818490">What's</span> <span m="818700">the</span> <span m="818800">expected</span> <span m="819290">number</span> <span m="819650">of</span> <span m="819890">men</span> <span m="820150">to</span> <span m="820220">get</span> <span m="820350">the</span> <span m="820440">right</span> <span m="820630">hat</span> <span m="820850">back?</span> </p>
<p><span m="824320">One,</span> <span m="824960">same</span> <span m="825270">as</span> <span m="825550">the</span> <span m="825670">other</span> <span m="825830">one.</span> <span m="826990">Because</span> <span m="827170">you've</span> <span m="827240">got</span> <span m="827370">n</span> <span m="827530">men</span> <span m="827760">each</span> <span m="827930">with</span> <span m="828030">a</span> <span m="828080">1</span> <span m="828175">in</span> <span m="828270">n</span> <span m="828540">chance,</span> <span m="828900">so it's</span> <span m="829080">1.</span> <span m="830130">Markov</span> <span m="830790">says</span> <span m="832150">the</span> <span m="832260">probability</span> <span m="832890">that</span> <span m="833110">n</span> <span m="833340">men</span> <span m="833920">get</span> <span m="834110">the</span> <span m="834190">right</span> <span m="834380">hat</span> <span m="834590">back</span> <span m="834850">is</span> <span m="834950">at</span> <span m="835040">most</span> <span m="835360">1</span> <span m="835515">in</span> <span m="835670">n,</span> <span m="836090">same</span> <span m="836380">as</span> <span m="836480">before.</span> <span m="838680">What's</span> <span m="838860">the</span> <span m="838980">actual</span> <span m="839350">probability</span> <span m="840560">that</span> <span m="840700">all</span> <span m="841090">n</span> <span m="841350">men</span> <span m="841590">get</span> <span m="841740">the</span> <span m="841820">right</span> <span m="842010">hat</span> <span m="842210">back?</span> </p>
<p><span m="843550">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="845070">PROFESSOR: 1</span> <span m="845430">in</span> <span m="845570">n</span> <span m="845770">factorial.</span> <span m="847550">So</span> <span m="847620">in</span> <span m="847700">this</span> <span m="847890">case,</span> <span m="848250">Markov</span> <span m="848800">is</span> <span m="848950">way</span> <span m="849480">off</span> <span m="849700">the</span> <span m="849800">mark.</span> <span m="850120">It</span> <span m="850220">says</span> <span m="850430">1</span> <span m="850550">in</span> <span m="850670">n.</span> <span m="851810">But</span> <span m="851950">in</span> <span m="852050">fact</span> <span m="852320">the</span> <span m="852410">real</span> <span m="852670">bound</span> <span m="852960">is</span> <span m="853080">much</span> <span m="853390">smaller.</span> <span m="855480">So</span> <span m="856300">Markov</span> <span m="856700">is</span> <span m="856790">not</span> <span m="857010">always</span> <span m="858520">tight.</span> <span m="859010">It's</span> <span m="859240">always</span> <span m="859600">an</span> <span m="859680">upper</span> <span m="859780">bound.</span> <span m="860230">But</span> <span m="860360">it</span> <span m="860500">sometimes</span> <span m="861640">is</span> <span m="861780">not</span> <span m="861980">the</span> <span m="862070">right</span> <span m="862280">answer.</span> <span m="863110">And</span> <span m="863260">to</span> <span m="863320">get</span> <span m="863430">the</span> <span m="863520">right</span> <span m="863730">answer,</span> <span m="864530">often</span> <span m="864960">you</span> <span m="865030">need</span> <span m="865210">to</span> <span m="865270">know</span> <span m="865450">more</span> <span m="865990">about</span> <span m="866460">the</span> <span m="866560">distribution.</span> </p>
<p><span m="870350">OK,</span> <span m="871880">what</span> <span m="872220">if</span> <span m="872570">R</span> <span m="872845">can</span> <span m="873120">be</span> <span m="873210">negative?</span> <span m="875580">Is</span> <span m="875730">it</span> <span m="875820">possible</span> <span m="876230">that</span> <span m="876360">Markov's</span> <span m="876820">theorem</span> <span m="877160">holds</span> <span m="877470">there?</span> <span m="880540">Because</span> <span m="880690">I</span> <span m="880750">use</span> <span m="881690">the</span> <span m="881770">assumption</span> <span m="883210">in</span> <span m="883310">the</span> <span m="883380">theorem.</span> <span m="884240">Can</span> <span m="884670">anybody</span> <span m="884970">give</span> <span m="885130">me</span> <span m="885210">an</span> <span m="885270">example</span> <span m="885750">where</span> <span m="885970">it</span> <span m="886040">doesn't</span> <span m="886340">work</span> <span m="886600">if</span> <span m="886920">R</span> <span m="887210">can</span> <span m="887560">be</span> <span m="887660">negative?</span> </p>
<p><span m="890545">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="893540">PROFESSOR: Yeah,</span> <span m="894230">good,</span> <span m="895210">so</span> <span m="895580">for</span> <span m="895710">example,</span> <span m="896320">say</span> <span m="897450">probability</span> <span m="898060">R</span> <span m="898460">equals</span> <span m="899010">1,000</span> <span m="901270">is</span> <span m="901500">1/2,</span> <span m="902660">and</span> <span m="902950">the</span> <span m="903020">probability</span> <span m="904350">R</span> <span m="904770">equals</span> <span m="905180">minus</span> <span m="905620">1,000</span> <span m="907090">is</span> <span m="907280">1/2.</span> <span m="908500">Then</span> <span m="908790">the</span> <span m="908890">expected</span> <span m="909430">value</span> <span m="909740">of</span> <span m="909840">R</span> <span m="910586">is</span> <span m="911320">0.</span> <span m="913440">And</span> <span m="913780">say</span> <span m="913980">we</span> <span m="914160">asked</span> <span m="914510">the</span> <span m="914580">probability</span> <span m="915300">that</span> <span m="915500">R</span> <span m="915880">is</span> <span m="916140">at</span> <span m="916240">least</span> <span m="916530">1,000.</span> <span m="918310">Well,</span> <span m="919500">that's</span> <span m="919800">going</span> <span m="919920">to</span> <span m="919990">be</span> <span m="920160">1/2.</span> <span m="921700">But</span> <span m="921890">that</span> <span m="922060">does</span> <span m="922240">not</span> <span m="922580">equal</span> <span m="923490">the</span> <span m="923600">expected</span> <span m="924150">value</span> <span m="924440">of</span> <span m="924550">R/1,000,</span> <span m="927730">which</span> <span m="927930">would</span> <span m="928040">be</span> <span m="928140">0.</span> </p>
<p><span m="930150">So</span> <span m="930460">Markov's</span> <span m="930720">theorem</span> <span m="931370">really</span> <span m="931690">does</span> <span m="932100">need</span> <span m="933970">that</span> <span m="934520">R</span> <span m="934970">to</span> <span m="935280">be</span> <span m="935390">non-negative.</span> <span m="936970">In fact,</span> <span m="937170">let's</span> <span m="937310">see</span> <span m="937420">if</span> <span m="937540">we</span> <span m="937630">saw</span> <span m="937900">where</span> <span m="938050">we</span> <span m="938180">used</span> <span m="938440">it in</span> <span m="938570">the</span> <span m="938650">proof.</span> <span m="940910">Anybody</span> <span m="941240">see</span> <span m="941450">where</span> <span m="941570">we</span> <span m="941700">use</span> <span m="941950">that</span> <span m="942170">fact</span> <span m="942440">in</span> <span m="942500">the</span> <span m="942570">proof,</span> <span m="942920">that</span> <span m="943050">R</span> <span m="943260">can't</span> <span m="943520">be</span> <span m="943610">negative?</span> <span m="946050">What</span> <span m="946200">is</span> <span m="946320">it?</span> </p>
<p><span m="947148">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="949340">PROFESSOR: Well,</span> <span m="949940">no,</span> <span m="950310">because</span> <span m="950500">x</span> <span m="950710">is</span> <span m="950800">positive.</span> <span m="952236">We</span> <span m="952660">said</span> <span m="952890">x</span> <span m="953080">is</span> <span m="953200">positive.</span> <span m="955080">So</span> <span m="955230">it's</span> <span m="955360">not</span> <span m="955600">used</span> <span m="955920">there.</span> <span m="956160">But</span> <span m="956250">that's</span> <span m="956400">a</span> <span m="956450">good</span> <span m="956620">one</span> <span m="956750">to</span> <span m="957240">look</span> <span m="957520">at.</span> <span m="958850">Yeah?</span> </p>
<p><span m="961320">AUDIENCE: [INAUDIBLE]</span> <span m="965180">is</span> <span m="965310">greater</span> <span m="965510">than or</span> <span m="965690">equal to 0.</span> </p>
<p><span m="966630">PROFESSOR: Yeah,</span> <span m="967580">if</span> <span m="967730">R</span> <span m="967800">can</span> <span m="968020">be</span> <span m="968270">negative,</span> <span m="970120">then</span> <span m="970270">this</span> <span m="970630">is</span> <span m="970770">not</span> <span m="971200">necessarily</span> <span m="971920">a</span> <span m="971970">positive</span> <span m="972420">number.</span> <span m="973270">It</span> <span m="973300">could</span> <span m="973410">be</span> <span m="973500">a</span> <span m="973600">negative</span> <span m="973950">number.</span> <span m="974710">And</span> <span m="974890">then</span> <span m="975240">this</span> <span m="975440">inequality</span> <span m="976010">doesn't</span> <span m="976760">hold.</span> <span m="980740">OK,</span> <span m="981320">good.</span> <span m="985530">All</span> <span m="985710">right,</span> <span m="986150">now</span> <span m="986920">it</span> <span m="987100">turns</span> <span m="987440">out</span> <span m="987810">there</span> <span m="987940">is</span> <span m="988070">a</span> <span m="988120">variation</span> <span m="988820">of</span> <span m="988890">Markov's</span> <span m="989390">theorem</span> <span m="989720">you</span> <span m="989820">can</span> <span m="989990">use</span> <span m="990890">when</span> <span m="991060">R</span> <span m="991240">is</span> <span m="991350">negative.</span> <span m="992050">Yeah.</span> </p>
<p><span m="993379">AUDIENCE: [INAUDIBLE]</span> <span m="994708">but</span> <span m="995151">would</span> <span m="995600">it</span> <span m="996080">be OK just to shift everything</span> <span m="996560">up?</span> </p>
<p><span m="997040">PROFESSOR: Yeah,</span> <span m="998160">yeah,</span> <span m="998670">that's</span> <span m="998730">great.</span> <span m="999880">If</span> <span m="1000050">R</span> <span m="1000140">has</span> <span m="1000290">a</span> <span m="1000350">limit</span> <span m="1000800">on</span> <span m="1000930">how</span> <span m="1001100">negative</span> <span m="1001530">it</span> <span m="1001630">can</span> <span m="1001760">be,</span> <span m="1003120">then</span> <span m="1003400">you</span> <span m="1003500">make</span> <span m="1003680">an</span> <span m="1003830">R</span> <span m="1004100">prime,</span> <span m="1004640">which</span> <span m="1004700">just</span> <span m="1005250">adds</span> <span m="1005640">that</span> <span m="1005870">limit</span> <span m="1006190">to</span> <span m="1006340">R,</span> <span m="1007430">makes</span> <span m="1007770">it</span> <span m="1007860">positive</span> <span m="1008145">or</span> <span m="1008430">non-negative.</span> <span m="1009710">And</span> <span m="1009870">now</span> <span m="1010020">use</span> <span m="1010210">Markov's</span> <span m="1010420">theorem</span> <span m="1010640">there.</span> <span m="1011890">And</span> <span m="1012030">that</span> <span m="1012250">is</span> <span m="1012570">now</span> <span m="1013380">an</span> <span m="1013510">analogous</span> <span m="1013990">form</span> <span m="1014350">of</span> <span m="1014450">Markov's</span> <span m="1014960">theorem</span> <span m="1015540">when</span> <span m="1015930">R</span> <span m="1016260">can</span> <span m="1016400">be</span> <span m="1016490">negative,</span> <span m="1016960">but</span> <span m="1017090">there's</span> <span m="1017240">a</span> <span m="1017300">lower</span> <span m="1017610">limit</span> <span m="1018270">to</span> <span m="1018460">it.</span> <span m="1019220">And</span> <span m="1019500">I</span> <span m="1019570">won't</span> <span m="1020470">stay to</span> <span m="1020880">improve</span> <span m="1021240">that</span> <span m="1021450">here.</span> <span m="1021625">But</span> <span m="1021800">that's</span> <span m="1022000">in</span> <span m="1022140">the</span> <span m="1022210">text</span> <span m="1022680">and</span> <span m="1022770">something</span> <span m="1023000">you</span> <span m="1023090">want</span> <span m="1023300">to</span> <span m="1023360">be</span> <span m="1023620">familiar</span> <span m="1024069">with.</span> </p>
<p><span m="1026770">What</span> <span m="1026950">I</span> <span m="1027020">do</span> <span m="1027200">want</span> <span m="1027400">to</span> <span m="1027460">do</span> <span m="1028069">in</span> <span m="1028220">class</span> <span m="1029359">is</span> <span m="1029540">another</span> <span m="1029849">case</span> <span m="1030319">where</span> <span m="1032119">you</span> <span m="1032300">can</span> <span m="1032480">use</span> <span m="1032680">Markov's</span> <span m="1033190">theorem</span> <span m="1033750">to</span> <span m="1034130">analyze</span> <span m="1034680">the</span> <span m="1034760">probability</span> <span m="1035400">or</span> <span m="1035500">upper</span> <span m="1035740">bound</span> <span m="1035940">the</span> <span m="1036010">probability</span> <span m="1036510">that</span> <span m="1036670">R</span> <span m="1036910">is</span> <span m="1037050">very</span> <span m="1037270">small,</span> <span m="1038500">less</span> <span m="1038770">than</span> <span m="1038869">its</span> <span m="1039020">expectation.</span> <span m="1039770">And</span> <span m="1039910">it's</span> <span m="1040000">the</span> <span m="1040069">same</span> <span m="1040329">idea</span> <span m="1040650">as</span> <span m="1040730">you</span> <span m="1040810">just</span> <span m="1041050">suggested.</span> <span m="1043020">So</span> <span m="1043630">let's</span> <span m="1043829">state</span> <span m="1044119">that.</span> </p>
<p><span m="1049610">If</span> <span m="1049910">R</span> <span m="1050180">is</span> <span m="1050440">upper</span> <span m="1050740">bounded,</span> <span m="1051290">has</span> <span m="1051510">a</span> <span m="1051570">hard</span> <span m="1051890">limit</span> <span m="1052140">on</span> <span m="1052230">the</span> <span m="1052330">upper</span> <span m="1052510">bound,</span> <span m="1052790">by</span> <span m="1052910">u</span> <span m="1056780">for</span> <span m="1057040">some</span> <span m="1058030">u</span> <span m="1058810">in</span> <span m="1058960">the</span> <span m="1059060">real</span> <span m="1059250">numbers,</span> <span m="1062460">then</span> <span m="1063370">for</span> <span m="1063630">all</span> <span m="1064180">x</span> <span m="1064640">less</span> <span m="1064990">than</span> <span m="1065170">u,</span> <span m="1068700">the</span> <span m="1068820">probability</span> <span m="1069570">that</span> <span m="1069770">R</span> <span m="1070210">is</span> <span m="1071880">less</span> <span m="1072140">than or</span> <span m="1072260">equal</span> <span m="1072490">to</span> <span m="1072600">x</span> <span m="1074660">is</span> <span m="1074890">at</span> <span m="1075010">most</span> <span m="1076240">u</span> <span m="1076490">minus</span> <span m="1076960">the</span> <span m="1077050">expected</span> <span m="1077570">value</span> <span m="1077870">of</span> <span m="1078310">R</span> <span m="1079810">over</span> <span m="1080820">u</span> <span m="1081130">minus</span> <span m="1081560">x.</span> </p>
<p><span m="1083960">So</span> <span m="1084250">in</span> <span m="1084360">this</span> <span m="1084580">case,</span> <span m="1085040">we're</span> <span m="1085140">getting</span> <span m="1085280">a</span> <span m="1085340">probability</span> <span m="1086150">that</span> <span m="1087440">R</span> <span m="1087930">is</span> <span m="1088290">less</span> <span m="1088590">than</span> <span m="1088730">something</span> <span m="1089270">instead</span> <span m="1089390">of</span> <span m="1089530">R</span> <span m="1089770">is</span> <span m="1089920">bigger</span> <span m="1090170">than</span> <span m="1090310">something.</span> <span m="1092440">And</span> <span m="1092570">we're</span> <span m="1092660">going</span> <span m="1092790">to</span> <span m="1092880">do</span> <span m="1093180">it</span> <span m="1097090">using</span> <span m="1097500">a</span> <span m="1097550">simple</span> <span m="1097920">trick</span> <span m="1098250">that</span> <span m="1098350">we'll</span> <span m="1098460">be</span> <span m="1098620">sort</span> <span m="1098850">of</span> <span m="1098920">using</span> <span m="1099230">all</span> <span m="1099410">day,</span> <span m="1099720">really.</span> <span m="1101700">The</span> <span m="1101810">probability</span> <span m="1102360">that</span> <span m="1102500">R</span> <span m="1102770">is</span> <span m="1102930">less</span> <span m="1103150">than</span> <span m="1103270">x,</span> <span m="1105900">this</span> <span m="1106190">event,</span> <span m="1106690">R</span> <span m="1106950">is</span> <span m="1107080">less</span> <span m="1107270">than</span> <span m="1107400">x,</span> <span m="1107710">is</span> <span m="1107860">the</span> <span m="1107960">same</span> <span m="1108340">as</span> <span m="1108470">the</span> <span m="1108610">event</span> <span m="1110370">u</span> <span m="1110640">minus</span> <span m="1111060">R</span> <span m="1112670">is</span> <span m="1112890">at</span> <span m="1113320">least</span> <span m="1114700">u</span> <span m="1114950">minus</span> <span m="1115340">x.</span> </p>
<p><span m="1118660">So</span> <span m="1118770">what</span> <span m="1118900">have</span> <span m="1119000">I</span> <span m="1119080">done?</span> <span m="1120160">I</span> <span m="1120870">put</span> <span m="1122030">negative</span> <span m="1122460">R</span> <span m="1122640">over</span> <span m="1122740">here,</span> <span m="1123000">subtract</span> <span m="1123530">x</span> <span m="1123780">from each</span> <span m="1123980">side,</span> <span m="1124740">add</span> <span m="1125090">u</span> <span m="1125280">to</span> <span m="1125410">each</span> <span m="1125660">side.</span> <span m="1128320">I've</span> <span m="1128380">got</span> <span m="1128540">to</span> <span m="1128600">put</span> <span m="1128720">a</span> <span m="1128770">less</span> <span m="1129040">than</span> <span m="1129230">or</span> <span m="1129360">equal</span> <span m="1129670">to</span> <span m="1129790">here.</span> <span m="1131890">So</span> <span m="1132590">R</span> <span m="1132895">is</span> <span m="1133200">less</span> <span m="1133410">than or</span> <span m="1133560">equal</span> <span m="1133780">to</span> <span m="1133850">x</span> <span m="1134310">if</span> <span m="1134530">and</span> <span m="1134620">only</span> <span m="1134950">if</span> <span m="1135200">u</span> <span m="1135410">minus</span> <span m="1135780">r</span> <span m="1136100">is</span> <span m="1136260">at</span> <span m="1136380">least</span> <span m="1136770">u</span> <span m="1136930">minus</span> <span m="1137260">x.</span> <span m="1138840">It's</span> <span m="1139320">simple</span> <span m="1140230">math</span> <span m="1140640">there.</span> </p>
<p><span m="1142510">And</span> <span m="1142670">now</span> <span m="1142870">I'm</span> <span m="1142950">going</span> <span m="1143080">to</span> <span m="1143300">apply</span> <span m="1143630">Markov</span> <span m="1144380">to</span> <span m="1144510">this.</span> <span m="1147250">I'm</span> <span m="1147320">going</span> <span m="1147470">to</span> <span m="1147530">apply</span> <span m="1147770">Markov</span> <span m="1148260">to</span> <span m="1148360">this</span> <span m="1148620">random</span> <span m="1148960">variable.</span> <span m="1150930">And</span> <span m="1151140">this</span> <span m="1151280">will</span> <span m="1151370">be</span> <span m="1151580">the</span> <span m="1151680">value</span> <span m="1152060">I</span> <span m="1152150">would</span> <span m="1152360">have</span> <span m="1152470">had</span> <span m="1152680">for</span> <span m="1152840">x up in</span> <span m="1153320">Markov's</span> <span m="1153770">theorem.</span> <span m="1155840">Why</span> <span m="1156130">is</span> <span m="1156240">it</span> <span m="1156330">OK</span> <span m="1156630">to</span> <span m="1156720">apply</span> <span m="1157000">Markov</span> <span m="1157540">to</span> <span m="1157670">u</span> <span m="1157840">minus</span> <span m="1158210">R?</span> </p>
<p><span m="1164150">AUDIENCE: You could</span> <span m="1164210">just</span> <span m="1164390">define</span> <span m="1164890">the new</span> <span m="1165230">random</span> <span m="1165300">variable</span> <span m="1165870">to</span> <span m="1165990">be</span> <span m="1166695">u</span> <span m="1166970">minus</span> <span m="1167420">R.</span> </p>
<p><span m="1168360">PROFESSOR: Yeah,</span> <span m="1169920">so</span> <span m="1170070">I</span> <span m="1170100">got</span> <span m="1170240">a</span> <span m="1170280">new</span> <span m="1170480">random</span> <span m="1170770">variable.</span> <span m="1171170">But</span> <span m="1171300">what</span> <span m="1171420">do</span> <span m="1171520">I</span> <span m="1171580">need</span> <span m="1171760">to</span> <span m="1171830">know</span> <span m="1172050">about</span> <span m="1172280">that</span> <span m="1172450">new</span> <span m="1172620">random</span> <span m="1172930">variable</span> <span m="1173280">to</span> <span m="1173340">apply</span> <span m="1173560">Markov?</span> </p>
<p><span m="1174270">AUDIENCE: u</span> <span m="1174660">is always greater than</span> <span m="1175106">R.</span> </p>
<p><span m="1176000">PROFESSOR: u</span> <span m="1176290">is always</span> <span m="1176540">greater</span> <span m="1176750">than</span> <span m="1176860">R,</span> <span m="1177450">or</span> <span m="1177950">at</span> <span m="1178020">least</span> <span m="1178220">as</span> <span m="1178300">big</span> <span m="1178430">as</span> <span m="1178500">R.</span> <span m="1178670">So</span> <span m="1178790">u</span> <span m="1179050">minus</span> <span m="1179460">R</span> <span m="1179730">is</span> <span m="1179910">always</span> <span m="1180240">non-negative.</span> <span m="1181650">So</span> <span m="1181860">I</span> <span m="1181920">can</span> <span m="1182030">apply</span> <span m="1182290">Markov</span> <span m="1182740">now.</span> <span m="1184360">And</span> <span m="1184510">when</span> <span m="1184700">I</span> <span m="1184760">apply</span> <span m="1185020">Markov,</span> <span m="1188670">I'll</span> <span m="1189160">get</span> <span m="1190130">this</span> <span m="1190360">is</span> <span m="1190460">at</span> <span m="1190590">most--</span> <span m="1192530">maybe</span> <span m="1192750">I'll</span> <span m="1192830">go</span> <span m="1192970">over</span> <span m="1193110">here.</span> <span m="1196680">The</span> <span m="1196760">probability</span> <span m="1197710">that--</span> <span m="1199180">ooh,</span> <span m="1199780">not</span> <span m="1200170">R</span> <span m="1200390">here.</span> <span m="1200660">This</span> <span m="1200820">is</span> <span m="1200920">probability.</span> </p>
<p><span m="1202840">The</span> <span m="1203340">probability</span> <span m="1204070">that</span> <span m="1204240">u</span> <span m="1204460">minus</span> <span m="1204870">R</span> <span m="1206860">is</span> <span m="1207390">at</span> <span m="1207540">least</span> <span m="1208390">u</span> <span m="1208670">minus</span> <span m="1209090">x</span> <span m="1210250">is</span> <span m="1210470">at</span> <span m="1210630">most</span> <span m="1211940">the</span> <span m="1212050">expected</span> <span m="1212720">value</span> <span m="1213090">of</span> <span m="1213160">that</span> <span m="1213400">random</span> <span m="1213710">variable</span> <span m="1217960">over</span> <span m="1219160">this</span> <span m="1219390">value.</span> <span m="1223210">And</span> <span m="1224100">well</span> <span m="1224230">now</span> <span m="1224460">I</span> <span m="1224550">just</span> <span m="1224780">use</span> <span m="1224960">the</span> <span m="1225040">linearity</span> <span m="1225450">of</span> <span m="1225540">expectation.</span> <span m="1226260">I've</span> <span m="1226310">got</span> <span m="1226440">a</span> <span m="1226470">scalar</span> <span m="1226940">here.</span> <span m="1227290">So</span> <span m="1227420">this</span> <span m="1227690">is</span> <span m="1227940">u</span> <span m="1228390">minus</span> <span m="1229170">the</span> <span m="1229440">expected</span> <span m="1229930">value</span> <span m="1230220">of</span> <span m="1230330">R</span> <span m="1231710">over</span> <span m="1232160">u</span> <span m="1232460">minus</span> <span m="1232850">x.</span> <span m="1236490">So I've</span> <span m="1236700">used</span> <span m="1236980">Markov's</span> <span m="1237340">theorem</span> <span m="1237760">to</span> <span m="1237820">get</span> <span m="1237990">a</span> <span m="1238650">different</span> <span m="1239070">version</span> <span m="1239500">of</span> <span m="1239620">it.</span> </p>
<p><span m="1242930">All</span> <span m="1243170">right,</span> <span m="1243270">let's</span> <span m="1243470">do</span> <span m="1243590">an</span> <span m="1243670">example.</span> <span m="1245480">Say</span> <span m="1245760">I'm</span> <span m="1245990">looking</span> <span m="1246300">at</span> <span m="1247340">test</span> <span m="1247620">scores.</span> <span m="1250460">And</span> <span m="1251010">I'll</span> <span m="1251180">let</span> <span m="1251390">R</span> <span m="1251790">be</span> <span m="1252100">the</span> <span m="1252640">score</span> <span m="1253280">of</span> <span m="1253360">a</span> <span m="1253460">random</span> <span m="1253840">student</span> <span m="1256860">uniformly</span> <span m="1257510">selected.</span> <span m="1263390">And</span> <span m="1263740">say</span> <span m="1264110">that</span> <span m="1264240">the</span> <span m="1264320">max</span> <span m="1264700">score</span> <span m="1267690">is</span> <span m="1267870">100.</span> <span m="1270920">So</span> <span m="1271080">that's</span> <span m="1271270">u.</span> <span m="1272746">All</span> <span m="1273240">scores</span> <span m="1273680">are</span> <span m="1273720">at</span> <span m="1273790">most</span> <span m="1274130">100.</span> </p>
<p><span m="1275230">And</span> <span m="1275430">say</span> <span m="1275690">that</span> <span m="1275820">I</span> <span m="1275900">tell</span> <span m="1276120">you</span> <span m="1276220">the</span> <span m="1276320">class</span> <span m="1276730">average,</span> <span m="1278530">or</span> <span m="1278690">the</span> <span m="1278800">expected</span> <span m="1279360">value</span> <span m="1279670">of</span> <span m="1279780">R,</span> <span m="1283520">is</span> <span m="1283730">75.</span> <span m="1287360">And</span> <span m="1287540">now</span> <span m="1287800">I</span> <span m="1287890">want</span> <span m="1288150">to</span> <span m="1288210">know,</span> <span m="1289720">what's</span> <span m="1290050">the</span> <span m="1290150">probability</span> <span m="1291930">that</span> <span m="1292070">a</span> <span m="1292170">random</span> <span m="1292520">student</span> <span m="1292930">scores</span> <span m="1294300">50</span> <span m="1294710">or</span> <span m="1294760">below?</span> <span m="1301750">Can we</span> <span m="1302080">figure</span> <span m="1302320">that</span> <span m="1302530">out?</span> <span m="1303860">I</span> <span m="1304000">don't</span> <span m="1304120">know</span> <span m="1304250">anything</span> <span m="1304470">about</span> <span m="1304660">the</span> <span m="1304720">distribution,</span> <span m="1306130">just</span> <span m="1306460">that</span> <span m="1306570">the</span> <span m="1306640">max</span> <span m="1307020">score</span> <span m="1307390">is</span> <span m="1307510">100</span> <span m="1308450">and</span> <span m="1308660">the</span> <span m="1308930">average</span> <span m="1309280">score</span> <span m="1309550">is</span> <span m="1309640">75.</span> <span m="1311160">What's</span> <span m="1311490">the</span> <span m="1311570">probability</span> <span m="1313920">that</span> <span m="1314190">a</span> <span m="1314400">random</span> <span m="1314720">student</span> <span m="1315080">scores</span> <span m="1315540">50</span> <span m="1315960">or</span> <span m="1316050">less?</span> <span m="1316440">I</span> <span m="1316510">want</span> <span m="1316630">to</span> <span m="1316690">upper</span> <span m="1316870">bound</span> <span m="1317180">that.</span> </p>
<p><span m="1319750">So</span> <span m="1321230">we</span> <span m="1321340">just</span> <span m="1322100">plug</span> <span m="1322220">it</span> <span m="1322390">into</span> <span m="1322580">the</span> <span m="1322650">formula.</span> <span m="1324860">u</span> <span m="1325240">is</span> <span m="1325340">100.</span> <span m="1327890">The</span> <span m="1327990">expected</span> <span m="1328530">value</span> <span m="1328850">is</span> <span m="1328940">75.</span> <span m="1331300">u</span> <span m="1331660">is</span> <span m="1331750">100.</span> <span m="1333430">And</span> <span m="1333990">x</span> <span m="1334410">is</span> <span m="1334580">50.</span> <span m="1336960">And</span> <span m="1337110">that's</span> <span m="1337410">25</span> <span m="1338330">over</span> <span m="1338580">50,</span> <span m="1340450">is</span> <span m="1340740">1/2.</span> <span m="1343560">So</span> <span m="1343830">at</span> <span m="1343970">most</span> <span m="1344550">half</span> <span m="1344940">the</span> <span m="1345050">class</span> <span m="1345580">can</span> <span m="1345700">score</span> <span m="1346020">50</span> <span m="1346440">or</span> <span m="1346490">below.</span> <span m="1348430">And</span> <span m="1348540">state</span> <span m="1348810">it</span> <span m="1348890">as</span> <span m="1348980">a</span> <span m="1349040">probability</span> <span m="1349650">question</span> <span m="1350080">or</span> <span m="1350250">deterministic</span> <span m="1350920">fact</span> <span m="1352160">if</span> <span m="1352310">I</span> <span m="1352380">know</span> <span m="1352580">the</span> <span m="1352720">average</span> <span m="1352970">is</span> <span m="1353050">75</span> <span m="1353340">and</span> <span m="1353630">the</span> <span m="1353730">max</span> <span m="1354030">is</span> <span m="1354110">100.</span> <span m="1355450">Of</span> <span m="1355510">course,</span> <span m="1355680">another</span> <span m="1355820">way</span> <span m="1355960">of</span> <span m="1356030">thinking</span> <span m="1356350">about</span> <span m="1356570">that</span> <span m="1356740">is</span> <span m="1356890">if</span> <span m="1357010">more</span> <span m="1357360">than</span> <span m="1357500">half</span> <span m="1357780">the</span> <span m="1357880">class</span> <span m="1358330">scored</span> <span m="1358680">50</span> <span m="1358980">or</span> <span m="1359030">below,</span> <span m="1359700">your</span> <span m="1359880">average</span> <span m="1360210">would</span> <span m="1360310">have</span> <span m="1360400">had</span> <span m="1360640">to</span> <span m="1360750">be</span> <span m="1360910">lower,</span> <span m="1362740">even</span> <span m="1362980">if</span> <span m="1363070">everybody</span> <span m="1363380">else</span> <span m="1363580">was</span> <span m="1363720">right</span> <span m="1363910">at</span> <span m="1363970">100.</span> <span m="1364430">It</span> <span m="1364530">wouldn't</span> <span m="1364790">average</span> <span m="1365040">out to</span> <span m="1365260">75.</span> <span m="1368980">All right,</span> <span m="1369200">any</span> <span m="1369350">questions</span> <span m="1371420">about</span> <span m="1371720">that?</span> </p>
<p><span m="1374410">OK,</span> <span m="1377850">so</span> <span m="1379370">sometimes</span> <span m="1379940">Markov</span> <span m="1380450">is</span> <span m="1380590">dead</span> <span m="1380840">on</span> <span m="1381040">right,</span> <span m="1381410">gives</span> <span m="1381650">the</span> <span m="1381750">right</span> <span m="1381960">answer.</span> <span m="1382740">For</span> <span m="1382860">example,</span> <span m="1383620">half</span> <span m="1383850">the</span> <span m="1383940">class</span> <span m="1384280">could</span> <span m="1384430">have</span> <span m="1384520">scored</span> <span m="1384790">50,</span> <span m="1385020">and</span> <span m="1385250">half</span> <span m="1385490">could</span> <span m="1385610">have</span> <span m="1385700">gotten</span> <span m="1385900">100</span> <span m="1386640">to</span> <span m="1386770">make</span> <span m="1386940">it</span> <span m="1387010">be</span> <span m="1387080">75.</span> <span m="1388250">And</span> <span m="1388390">sometimes</span> <span m="1388800">it's</span> <span m="1388910">way</span> <span m="1389150">off,</span> <span m="1389920">like</span> <span m="1390120">in</span> <span m="1390190">the</span> <span m="1390260">hat</span> <span m="1390460">check</span> <span m="1390670">problem.</span> </p>
<p><span m="1392240">Now,</span> <span m="1392670">if</span> <span m="1392790">you</span> <span m="1392900">know</span> <span m="1393150">more</span> <span m="1393370">about</span> <span m="1393590">the</span> <span m="1393650">distribution,</span> <span m="1395270">then</span> <span m="1395570">you</span> <span m="1395680">can</span> <span m="1395830">get</span> <span m="1395990">better</span> <span m="1396200">bounds,</span> <span m="1397230">especially</span> <span m="1397770">the</span> <span m="1397840">cases</span> <span m="1398190">when</span> <span m="1398330">you're</span> <span m="1398710">far</span> <span m="1399000">off.</span> <span m="1400430">For</span> <span m="1400560">example,</span> <span m="1401060">if</span> <span m="1401160">you</span> <span m="1401260">know</span> <span m="1401330">the</span> <span m="1401420">variance</span> <span m="1402340">in</span> <span m="1402470">addition</span> <span m="1402960">to</span> <span m="1403070">the</span> <span m="1403160">expectation,</span> <span m="1404230">or</span> <span m="1404440">aside</span> <span m="1404750">from</span> <span m="1404850">the</span> <span m="1404920">expectation,</span> <span m="1406510">then</span> <span m="1406790">you</span> <span m="1406880">can</span> <span m="1407020">get</span> <span m="1407170">better</span> <span m="1407400">bounds</span> <span m="1408210">on</span> <span m="1408380">the</span> <span m="1408450">probability</span> <span m="1408970">that</span> <span m="1409090">the</span> <span m="1409190">random</span> <span m="1409470">variable</span> <span m="1409840">is</span> <span m="1409960">large.</span> </p>
<p><span m="1411220">And</span> <span m="1411850">in</span> <span m="1411970">this</span> <span m="1412120">case,</span> <span m="1412960">the</span> <span m="1412980">result</span> <span m="1413410">is</span> <span m="1413550">known</span> <span m="1413780">as</span> <span m="1414060">Chebyshev's</span> <span m="1414620">theorem.</span> <span m="1416980">I'll</span> <span m="1417110">do</span> <span m="1417260">that</span> <span m="1417420">over</span> <span m="1417590">here.</span> <span m="1420840">And</span> <span m="1421020">it's</span> <span m="1421180">the</span> <span m="1421580">analog</span> <span m="1422250">of</span> <span m="1422350">Markov's</span> <span m="1422880">theorem</span> <span m="1425140">based</span> <span m="1425430">on</span> <span m="1425580">variance.</span> <span m="1429260">It</span> <span m="1430620">says,</span> <span m="1431010">for</span> <span m="1431190">all</span> <span m="1431570">x</span> <span m="1432050">bigger</span> <span m="1432310">than</span> <span m="1432480">0,</span> <span m="1434170">and</span> <span m="1435660">any</span> <span m="1436100">random</span> <span m="1436500">variable</span> <span m="1436950">R--</span> <span m="1437710">could</span> <span m="1437930">even</span> <span m="1438120">be</span> <span m="1438210">negative--</span> <span m="1441230">the</span> <span m="1441370">probability</span> <span m="1443180">that</span> <span m="1443370">R</span> <span m="1443746">deviates</span> <span m="1444500">from</span> <span m="1444670">its</span> <span m="1444810">expected</span> <span m="1445300">value</span> <span m="1446010">in</span> <span m="1446200">either</span> <span m="1446440">direction</span> <span m="1448720">by</span> <span m="1449140">at</span> <span m="1449240">least</span> <span m="1449590">x</span> <span m="1451340">is</span> <span m="1451590">at</span> <span m="1451720">most</span> <span m="1452120">of</span> <span m="1452180">the</span> <span m="1452240">variance</span> <span m="1452930">of</span> <span m="1453050">R</span> <span m="1455150">divided</span> <span m="1455590">by</span> <span m="1455780">x</span> <span m="1456030">squared.</span> </p>
<p><span m="1458850">So</span> <span m="1459150">this</span> <span m="1459390">is</span> <span m="1459510">like</span> <span m="1459760">Markov's</span> <span m="1460030">theorem,</span> <span m="1460610">except</span> <span m="1461060">that</span> <span m="1461430">we're</span> <span m="1462330">now</span> <span m="1462690">bounding</span> <span m="1463280">the</span> <span m="1463340">deviation</span> <span m="1464640">in</span> <span m="1464770">either</span> <span m="1464950">direction.</span> <span m="1466140">Instead</span> <span m="1466500">of</span> <span m="1466600">expected</span> <span m="1467080">value, you have</span> <span m="1467530">variance.</span> <span m="1468270">Instead</span> <span m="1468540">of x,</span> <span m="1468630">you've</span> <span m="1468870">got</span> <span m="1469000">x</span> <span m="1469140">squared,</span> <span m="1470480">but</span> <span m="1470620">the</span> <span m="1470690">same</span> <span m="1471190">idea.</span> <span m="1472930">In</span> <span m="1473020">fact,</span> <span m="1473890">the</span> <span m="1474020">proof</span> <span m="1474390">uses</span> <span m="1474700">Markov's</span> <span m="1475170">theorem.</span> </p>
<p><span m="1483330">Well,</span> <span m="1483530">the</span> <span m="1483610">probability</span> <span m="1485930">that</span> <span m="1486120">R</span> <span m="1486500">deviates</span> <span m="1487260">from</span> <span m="1487440">its</span> <span m="1487570">expected</span> <span m="1488040">value</span> <span m="1489540">by</span> <span m="1489780">at</span> <span m="1489880">least</span> <span m="1490210">x,</span> <span m="1492820">this</span> <span m="1493150">is</span> <span m="1493290">the</span> <span m="1493390">same</span> <span m="1493950">event,</span> <span m="1494450">or</span> <span m="1494640">happens</span> <span m="1495030">if</span> <span m="1495190">and</span> <span m="1495280">only</span> <span m="1495610">if</span> <span m="1496350">R</span> <span m="1496730">minus</span> <span m="1497170">expected</span> <span m="1497620">value</span> <span m="1497970">squared</span> <span m="1499430">is</span> <span m="1499620">at</span> <span m="1499740">least</span> <span m="1500050">x</span> <span m="1500240">squared.</span> <span m="1501450">I'm</span> <span m="1501550">just</span> <span m="1501690">going</span> <span m="1501810">to</span> <span m="1501880">square</span> <span m="1502390">both</span> <span m="1502630">sides</span> <span m="1503010">here.</span> <span m="1514960">OK,</span> <span m="1515280">I</span> <span m="1515360">square</span> <span m="1515760">both</span> <span m="1515970">sides.</span> <span m="1516310">And</span> <span m="1516400">since</span> <span m="1516560">this</span> <span m="1516710">is</span> <span m="1516830">positive</span> <span m="1517410">and</span> <span m="1517580">this</span> <span m="1517760">is</span> <span m="1517850">positive,</span> <span m="1518290">I</span> <span m="1518360">can</span> <span m="1518490">square</span> <span m="1518750">both</span> <span m="1518960">sides</span> <span m="1519390">and</span> <span m="1519490">maintain</span> <span m="1519810">the</span> <span m="1519900">inequality.</span> </p>
<p><span m="1523650">Now</span> <span m="1524680">I'm</span> <span m="1524860">going</span> <span m="1524990">to</span> <span m="1525060">apply</span> <span m="1526540">Markov's</span> <span m="1527270">theorem</span> <span m="1528050">to</span> <span m="1528210">that</span> <span m="1528510">random</span> <span m="1528880">variable.</span> <span m="1530840">It's a</span> <span m="1530990">random</span> <span m="1531270">variable. It's</span> <span m="1531670">R</span> <span m="1531970">minus</span> <span m="1532320">expected</span> <span m="1532740">value</span> <span m="1533040">squared.</span> <span m="1533520">So</span> <span m="1533620">it's</span> <span m="1533720">a</span> <span m="1533790">random</span> <span m="1534100">variable.</span> <span m="1535420">And</span> <span m="1535550">what's</span> <span m="1535750">nice</span> <span m="1536060">about</span> <span m="1536340">this</span> <span m="1536520">random</span> <span m="1536800">variable</span> <span m="1537180">that</span> <span m="1537310">lets</span> <span m="1537510">me</span> <span m="1537610">apply</span> <span m="1537890">Markov's</span> <span m="1538390">theorem?</span> <span m="1540920">It's</span> <span m="1541080">a</span> <span m="1541140">square.</span> <span m="1541820">So</span> <span m="1542030">it's</span> <span m="1542160">always</span> <span m="1542670">non-negative.</span> </p>
<p><span m="1543990">So</span> <span m="1544160">I</span> <span m="1544220">can</span> <span m="1544370">apply</span> <span m="1544740">Markov's</span> <span m="1545240">theorem.</span> <span m="1546910">And</span> <span m="1547040">my</span> <span m="1547160">Markov's</span> <span m="1547700">theorem,</span> <span m="1549110">this</span> <span m="1549280">probability</span> <span m="1549860">is</span> <span m="1549960">at</span> <span m="1550060">most</span> <span m="1550930">the</span> <span m="1550960">expected</span> <span m="1551520">value</span> <span m="1551880">of</span> <span m="1551960">that</span> <span m="1560710">divided</span> <span m="1561250">by</span> <span m="1561890">this.</span> <span m="1564180">That's</span> <span m="1564250">what</span> <span m="1564360">Markov's</span> <span m="1564810">theorem</span> <span m="1565100">says</span> <span m="1565930">as</span> <span m="1566050">long</span> <span m="1566270">as</span> <span m="1566400">this</span> <span m="1566590">is</span> <span m="1566700">always</span> <span m="1567570">non-negative.</span> <span m="1570030">All</span> <span m="1570210">right,</span> <span m="1570790">what's</span> <span m="1571210">a</span> <span m="1571280">simpler</span> <span m="1571770">expression</span> <span m="1572430">for</span> <span m="1572560">this,</span> <span m="1575080">the</span> <span m="1575210">expected</span> <span m="1575870">value</span> <span m="1576470">of</span> <span m="1576560">the</span> <span m="1577090">square</span> <span m="1577910">of</span> <span m="1578320">the</span> <span m="1578400">deviation</span> <span m="1579070">of</span> <span m="1579150">a</span> <span m="1579220">random</span> <span m="1579500">variable?</span> <span m="1581380">That's</span> <span m="1581690">the</span> <span m="1581790">variance.</span> <span m="1582370">That's</span> <span m="1582570">the</span> <span m="1582640">definition</span> <span m="1583170">of</span> <span m="1583250">variance.</span> </p>
<p><span m="1586470">So</span> <span m="1586680">that</span> <span m="1586950">is</span> <span m="1587080">just</span> <span m="1587420">the</span> <span m="1587490">variance</span> <span m="1589030">of</span> <span m="1589190">R</span> <span m="1590130">over</span> <span m="1590550">x</span> <span m="1590780">squared.</span> <span m="1591760">And</span> <span m="1591980">we're</span> <span m="1592070">done.</span> <span m="1593920">So</span> <span m="1594250">Chebyshev's</span> <span m="1594800">theorem</span> <span m="1596060">is</span> <span m="1596270">really</span> <span m="1596550">just</span> <span m="1596880">another</span> <span m="1597260">version</span> <span m="1597590">of</span> <span m="1597660">Markov's</span> <span m="1597920">theorem.</span> <span m="1599920">But</span> <span m="1600100">now</span> <span m="1600390">it's</span> <span m="1600530">based</span> <span m="1600820">on</span> <span m="1600930">the</span> <span m="1600990">variance.</span> <span m="1604450">OK,</span> <span m="1605150">any</span> <span m="1605770">questions?</span> </p>
<p><span m="1609150">OK,</span> <span m="1609900">so</span> <span m="1610120">there's</span> <span m="1610290">a</span> <span m="1610330">nice</span> <span m="1610560">corollary</span> <span m="1611970">for</span> <span m="1612070">this,</span> <span m="1613920">just</span> <span m="1614220">as</span> <span m="1614350">with</span> <span m="1614500">Markov's</span> <span m="1614970">theorem.</span> <span m="1616615">It</span> <span m="1617070">says</span> <span m="1617410">the</span> <span m="1617490">probability</span> <span m="1619150">that</span> <span m="1619310">the</span> <span m="1619440">absolute</span> <span m="1619870">value, the</span> <span m="1620290">deviation,</span> <span m="1625020">is</span> <span m="1625910">at</span> <span m="1626150">least</span> <span m="1627340">c</span> <span m="1628150">times</span> <span m="1628530">the</span> <span m="1628620">standard</span> <span m="1629240">deviation</span> <span m="1630460">of</span> <span m="1630650">R.</span> <span m="1632940">So</span> <span m="1633040">I'm</span> <span m="1633510">looking</span> <span m="1633730">at</span> <span m="1633790">the</span> <span m="1633870">probability</span> <span m="1634440">that</span> <span m="1634580">R</span> <span m="1634920">differs</span> <span m="1635310">from</span> <span m="1635450">its</span> <span m="1635600">expectation</span> <span m="1636880">by</span> <span m="1637090">at</span> <span m="1637190">least</span> <span m="1637720">some</span> <span m="1638010">scalar</span> <span m="1638440">c</span> <span m="1638960">times</span> <span m="1639490">the</span> <span m="1639560">standard</span> <span m="1639920">deviation.</span> </p>
<p><span m="1642760">Well,</span> <span m="1642900">what's</span> <span m="1643100">that?</span> <span m="1645290">Well,</span> <span m="1645500">that's</span> <span m="1645740">the</span> <span m="1645830">variance</span> <span m="1648360">of</span> <span m="1648550">R</span> <span m="1650020">over</span> <span m="1650680">the</span> <span m="1650790">square</span> <span m="1651550">of</span> <span m="1652580">this</span> <span m="1652820">thing--</span> <span m="1654490">c</span> <span m="1654800">squared</span> <span m="1656040">times</span> <span m="1656530">the</span> <span m="1657010">standard</span> <span m="1657440">deviation</span> <span m="1658040">squared.</span> <span m="1660590">What's</span> <span m="1660840">the</span> <span m="1660920">square</span> <span m="1661280">of</span> <span m="1661350">the</span> <span m="1661410">standard</span> <span m="1661740">deviation?</span> <span m="1663930">That's</span> <span m="1664170">the</span> <span m="1664240">variance.</span> <span m="1665930">They</span> <span m="1666130">cancel,</span> <span m="1666800">so</span> <span m="1666970">it's</span> <span m="1667060">just</span> <span m="1667230">1</span> <span m="1667400">over c</span> <span m="1667780">squared.</span> <span m="1671460">So</span> <span m="1672020">the</span> <span m="1672450">probability</span> <span m="1673160">of</span> <span m="1673230">more</span> <span m="1673490">than</span> <span m="1676730">twice</span> <span m="1677080">the</span> <span m="1677150">standard</span> <span m="1677410">deviation</span> <span m="1677840">off</span> <span m="1677930">the</span> <span m="1678010">expectation</span> <span m="1678340">is</span> <span m="1678670">at</span> <span m="1678750">most</span> <span m="1679020">1/4,</span> <span m="1680340">for</span> <span m="1680460">example.</span> <span m="1683780">All right,</span> <span m="1683950">let's</span> <span m="1684130">do</span> <span m="1684260">some</span> <span m="1684430">examples</span> <span m="1685090">of</span> <span m="1685160">that.</span> <span m="1688960">Maybe</span> <span m="1689080">we'll leave</span> <span m="1691050">Markov</span> <span m="1691570">up</span> <span m="1691750">there.</span> </p>
<p><span m="1702060">OK,</span> <span m="1702390">say</span> <span m="1702830">we're</span> <span m="1703060">looking</span> <span m="1703380">at</span> <span m="1703780">IQs.</span> <span m="1707540">In</span> <span m="1707920">this</span> <span m="1708060">case,</span> <span m="1708310">we're going to</span> <span m="1708440">let</span> <span m="1708660">R</span> <span m="1709210">be</span> <span m="1709400">the</span> <span m="1709600">IQ</span> <span m="1711430">of</span> <span m="1712060">a</span> <span m="1712160">random</span> <span m="1712480">person.</span> <span m="1718340">All</span> <span m="1718550">right,</span> <span m="1718730">now</span> <span m="1719100">we're</span> <span m="1719250">going</span> <span m="1719470">to</span> <span m="1719550">assume--</span> <span m="1721290">and</span> <span m="1721510">this</span> <span m="1721790">actually</span> <span m="1722230">is</span> <span m="1722440">the</span> <span m="1722520">case--</span> <span m="1725860">that</span> <span m="1726440">R</span> <span m="1726820">is</span> <span m="1727000">always</span> <span m="1727530">at</span> <span m="1727640">least</span> <span m="1727850">0,</span> <span m="1729580">despite</span> <span m="1730010">the</span> <span m="1730080">fact</span> <span m="1730430">that</span> <span m="1730720">probably</span> <span m="1731120">most</span> <span m="1731380">of</span> <span m="1731480">you</span> <span m="1732170">have</span> <span m="1732390">somebody</span> <span m="1732710">you</span> <span m="1732780">know</span> <span m="1732960">who</span> <span m="1733080">you</span> <span m="1733190">think</span> <span m="1733500">has</span> <span m="1733670">a</span> <span m="1733710">negative</span> <span m="1734130">IQ.</span> <span m="1735540">They</span> <span m="1735690">can't</span> <span m="1736040">be</span> <span m="1736140">negative.</span> <span m="1736500">They</span> <span m="1736650">have</span> <span m="1736850">to</span> <span m="1736930">be</span> <span m="1737650">non-zero.</span> </p>
<p><span m="1739000">In</span> <span m="1739170">fact,</span> <span m="1741220">IQs</span> <span m="1741930">are</span> <span m="1742220">adjusted.</span> <span m="1742970">So</span> <span m="1743120">the</span> <span m="1743240">expected</span> <span m="1743910">IQ</span> <span m="1745300">is</span> <span m="1745450">supposed</span> <span m="1745880">to</span> <span m="1745940">be</span> <span m="1746050">100,</span> <span m="1747260">although</span> <span m="1748670">actually</span> <span m="1749020">the</span> <span m="1749430">averages</span> <span m="1749750">may be</span> <span m="1750010">in</span> <span m="1750110">the</span> <span m="1750170">90's.</span> <span m="1751170">And</span> <span m="1751630">it's</span> <span m="1751730">set</span> <span m="1752020">up</span> <span m="1752180">so</span> <span m="1752410">that</span> <span m="1752590">the</span> <span m="1752690">standard</span> <span m="1753100">deviation</span> <span m="1753930">of</span> <span m="1754120">IQ</span> <span m="1755000">is</span> <span m="1755180">supposed</span> <span m="1755580">to</span> <span m="1755640">be</span> <span m="1755750">15.</span> <span m="1757690">So</span> <span m="1757850">we're</span> <span m="1757990">just</span> <span m="1758130">going</span> <span m="1758250">to</span> <span m="1758320">assume</span> <span m="1758620">those</span> <span m="1758880">are</span> <span m="1758990">facts</span> <span m="1760080">on</span> <span m="1760320">IQ.</span> <span m="1760710">And</span> <span m="1761200">that's</span> <span m="1761410">what</span> <span m="1761500">it's</span> <span m="1761620">meant</span> <span m="1761850">to</span> <span m="1761940">be.</span> </p>
<p><span m="1763260">And</span> <span m="1763410">now</span> <span m="1763570">we</span> <span m="1763670">want</span> <span m="1763880">to</span> <span m="1763940">know,</span> <span m="1765680">what's</span> <span m="1765920">the</span> <span m="1766010">probability</span> <span m="1766570">a</span> <span m="1766670">random</span> <span m="1766990">person</span> <span m="1767400">has</span> <span m="1767600">an</span> <span m="1767690">IQ</span> <span m="1768100">of</span> <span m="1768190">at</span> <span m="1768290">least</span> <span m="1768540">250?</span> <span m="1771050">Now</span> <span m="1771280">Marilyn,</span> <span m="1772100">from</span> <span m="1772260">"Ask</span> <span m="1772530">Marilyn,"</span> <span m="1773370">has</span> <span m="1773580">an</span> <span m="1773650">IQ</span> <span m="1774080">pretty</span> <span m="1774300">close</span> <span m="1774630">to</span> <span m="1774880">250.</span> <span m="1776070">And</span> <span m="1776390">she</span> <span m="1776500">thinks</span> <span m="1776760">that's</span> <span m="1777090">pretty</span> <span m="1777400">special,</span> <span m="1778770">pretty</span> <span m="1779020">rare.</span> <span m="1780700">So</span> <span m="1781010">what</span> <span m="1781190">can</span> <span m="1781310">we</span> <span m="1781410">say</span> <span m="1781690">about</span> <span m="1781980">that?</span> <span m="1782670">In</span> <span m="1782790">particular,</span> <span m="1784340">say</span> <span m="1784560">we</span> <span m="1784690">used</span> <span m="1784900">Markov.</span> <span m="1786740">What</span> <span m="1786970">could</span> <span m="1787090">you</span> <span m="1787200">say</span> <span m="1787510">about</span> <span m="1787720">the</span> <span m="1787800">probability</span> <span m="1789430">of</span> <span m="1790400">having</span> <span m="1790690">an</span> <span m="1790770">IQ</span> <span m="1791140">of</span> <span m="1791230">at</span> <span m="1791370">least</span> <span m="1791690">250?</span> <span m="1796800">What</span> <span m="1796930">does</span> <span m="1797040">Markov</span> <span m="1797490">tell</span> <span m="1797740">us?</span> </p>
<p><span m="1805858">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="1807340">PROFESSOR: What is</span> <span m="1807450">it?</span> </p>
<p><span m="1807997">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="1809160">PROFESSOR: Not</span> <span m="1809510">quite</span> <span m="1809740">1</span> <span m="1809890">in</span> <span m="1809960">25,</span> <span m="1810500">but</span> <span m="1810640">you're</span> <span m="1810740">on</span> <span m="1810840">the</span> <span m="1810940">right</span> <span m="1811150">track.</span> <span m="1814320">It's</span> <span m="1815620">not</span> <span m="1815830">quite</span> <span m="1816050">2/3.</span> <span m="1816790">It's</span> <span m="1817070">the</span> <span m="1817190">expected</span> <span m="1817790">value,</span> <span m="1820020">which</span> <span m="1820080">is</span> <span m="1820200">100,</span> <span m="1822080">over</span> <span m="1823920">the</span> <span m="1824040">x</span> <span m="1824260">value,</span> <span m="1824570">which</span> <span m="1824720">is</span> <span m="1824820">250.</span> <span m="1826930">So</span> <span m="1827020">it's</span> <span m="1827130">1</span> <span m="1827400">in</span> <span m="1828320">2.5,</span> <span m="1830032">or</span> <span m="1830460">0.4.</span> <span m="1832740">So</span> <span m="1833400">the</span> <span m="1833490">probability</span> <span m="1834040">is</span> <span m="1834150">at</span> <span m="1834800">most</span> <span m="1835200">0.4,</span> <span m="1835800">so</span> <span m="1835940">40%</span> <span m="1836660">chance</span> <span m="1837180">it</span> <span m="1837260">could</span> <span m="1837380">happen,</span> <span m="1838320">potentially,</span> <span m="1838850">but</span> <span m="1839040">no</span> <span m="1839320">bigger</span> <span m="1839830">than</span> <span m="1839960">that.</span> <span m="1841720">What</span> <span m="1841850">about</span> <span m="1842050">Chebyshev?</span> <span m="1845090">See</span> <span m="1845200">if</span> <span m="1845270">you</span> <span m="1845370">can</span> <span m="1845490">figure</span> <span m="1845760">out</span> <span m="1845900">what</span> <span m="1846040">Chebyshev</span> <span m="1846550">says</span> <span m="1846890">about</span> <span m="1847130">the</span> <span m="1847200">probability</span> <span m="1848580">of</span> <span m="1848720">having</span> <span m="1848980">an</span> <span m="1849070">IQ</span> <span m="1849490">of</span> <span m="1849580">at</span> <span m="1849670">least</span> <span m="1849890">250.</span> </p>
<p><span m="1859250">It's</span> <span m="1859440">a</span> <span m="1859480">little</span> <span m="1859660">tricky.</span> <span m="1859990">You've got</span> <span m="1860120">to</span> <span m="1860200">sort</span> <span m="1860360">of</span> <span m="1860430">plug</span> <span m="1860720">it</span> <span m="1860800">into</span> <span m="1861080">the</span> <span m="1861220">equation</span> <span m="1861790">there</span> <span m="1862070">and</span> <span m="1862940">get</span> <span m="1863100">it</span> <span m="1863180">to</span> <span m="1863260">fit</span> <span m="1863640">in</span> <span m="1863710">the</span> <span m="1863800">right</span> <span m="1864010">form.</span> <span m="1869240">Chebyshev</span> <span m="1871380">says</span> <span m="1872000">that--</span> <span m="1873990">let's</span> <span m="1874210">get</span> <span m="1874340">in</span> <span m="1874410">the</span> <span m="1874500">right</span> <span m="1874710">form.</span> <span m="1875870">I've</span> <span m="1876010">got</span> <span m="1876380">probability</span> <span m="1876780">that R</span> <span m="1877150">is at</span> <span m="1877230">least</span> <span m="1877470">250.</span> <span m="1878800">I've</span> <span m="1878900">got</span> <span m="1879080">to</span> <span m="1879390">get</span> <span m="1879730">it</span> <span m="1879840">into</span> <span m="1880020">that</span> <span m="1880380">form</span> <span m="1880780">up</span> <span m="1880960">there.</span> <span m="1883670">So</span> <span m="1885030">that's</span> <span m="1885280">the</span> <span m="1885360">probability</span> <span m="1886100">that--</span> <span m="1887630">well,</span> <span m="1887800">first</span> <span m="1888200">R</span> <span m="1888440">minus</span> <span m="1888760">100</span> <span m="1889480">is</span> <span m="1889610">at</span> <span m="1889740">least</span> <span m="1890180">150.</span> <span m="1892500">So</span> <span m="1892580">I've</span> <span m="1892680">got</span> <span m="1892890">R</span> <span m="1893140">minus</span> <span m="1893510">the</span> <span m="1893600">expected</span> <span m="1894160">value.</span> <span m="1894550">I'm</span> <span m="1894670">sort</span> <span m="1894880">of</span> <span m="1895070">getting</span> <span m="1895330">it</span> <span m="1895450">ready</span> <span m="1895690">to</span> <span m="1895770">apply</span> <span m="1896080">Chebyshev</span> <span m="1896680">here.</span> </p>
<p><span m="1898950">And</span> <span m="1899230">then</span> <span m="1899360">150--</span> <span m="1900580">how</span> <span m="1900730">many</span> <span m="1900890">standard</span> <span m="1901250">deviations</span> <span m="1901870">is</span> <span m="1901980">150?</span> <span m="1904630">10,</span> <span m="1905630">all</span> <span m="1905800">right?</span> <span m="1906440">So</span> <span m="1906880">this</span> <span m="1907140">is</span> <span m="1908660">the</span> <span m="1908740">probability</span> <span m="1909210">that</span> <span m="1909370">R</span> <span m="1909740">minus</span> <span m="1911080">the</span> <span m="1911170">expected</span> <span m="1911660">value</span> <span m="1911970">of</span> <span m="1912090">R</span> <span m="1913580">is</span> <span m="1913860">at</span> <span m="1913970">least</span> <span m="1914500">10</span> <span m="1915400">standard</span> <span m="1915840">deviations.</span> <span m="1918620">That's</span> <span m="1918780">what</span> <span m="1918910">I'm</span> <span m="1919020">asking.</span> <span m="1919480">I'm</span> <span m="1919630">not</span> <span m="1919920">quite</span> <span m="1920330">there.</span> <span m="1920690">I'm</span> <span m="1920790">going</span> <span m="1920910">to</span> <span m="1921000">use</span> <span m="1921190">the</span> <span m="1921260">corollary</span> <span m="1921830">there.</span> <span m="1921950">But</span> <span m="1922060">I've</span> <span m="1922120">got</span> <span m="1922240">to</span> <span m="1922300">get</span> <span m="1922430">that</span> <span m="1922590">absolute</span> <span m="1923030">value</span> <span m="1923450">thing</span> <span m="1923750">in.</span> </p>
<p><span m="1925110">But</span> <span m="1926420">it's</span> <span m="1926640">upper</span> <span m="1926890">bounded</span> <span m="1928320">by</span> <span m="1928480">the</span> <span m="1928570">probability</span> <span m="1929130">of</span> <span m="1929190">the</span> <span m="1929320">absolute</span> <span m="1929720">value</span> <span m="1930620">of</span> <span m="1930790">R</span> <span m="1931010">minus</span> <span m="1931550">expected</span> <span m="1931980">value</span> <span m="1934740">bigger</span> <span m="1934860">than</span> <span m="1934970">or</span> <span m="1935050">equal</span> <span m="1935270">to</span> <span m="1935360">10</span> <span m="1935830">standard</span> <span m="1936170">deviations.</span> <span m="1938920">Because</span> <span m="1940160">this</span> <span m="1940430">allows</span> <span m="1940840">for</span> <span m="1940960">two</span> <span m="1941130">cases.</span> <span m="1942730">R</span> <span m="1943130">is</span> <span m="1943290">10</span> <span m="1943560">standard</span> <span m="1943840">deviations</span> <span m="1944360">high,</span> <span m="1945480">and</span> <span m="1945630">R is</span> <span m="1945980">10</span> <span m="1946230">standard</span> <span m="1946530">deviations</span> <span m="1947010">low</span> <span m="1947670">or</span> <span m="1947820">more.</span> <span m="1949170">So</span> <span m="1949520">this</span> <span m="1949940">is</span> <span m="1950080">upper</span> <span m="1950290">bounded</span> <span m="1950760">by</span> <span m="1950930">that.</span> </p>
<p><span m="1951820">And</span> <span m="1952020">now</span> <span m="1952340">I</span> <span m="1952420">can</span> <span m="1952590">plug</span> <span m="1952945">in</span> <span m="1953870">Chebyshev</span> <span m="1954520">in</span> <span m="1954580">the</span> <span m="1954640">corollary</span> <span m="1955140">form.</span> <span m="1955430">And</span> <span m="1955530">what's</span> <span m="1955710">the</span> <span m="1955850">answer</span> <span m="1956130">when</span> <span m="1956250">I</span> <span m="1956310">do</span> <span m="1956450">that?</span> <span m="1960880">1</span> <span m="1961095">in</span> <span m="1961310">100--</span> <span m="1962210">the</span> <span m="1962640">probability</span> <span m="1962950">of</span> <span m="1963260">being</span> <span m="1963490">off</span> <span m="1963740">by</span> <span m="1963860">10</span> <span m="1964140">standard</span> <span m="1964480">deviations</span> <span m="1964980">or</span> <span m="1965020">more</span> <span m="1965300">is</span> <span m="1965390">at</span> <span m="1965500">most</span> <span m="1965760">1 in</span> <span m="1966175">100,</span> <span m="1966590">1 in</span> <span m="1966680">10</span> <span m="1966880">squared.</span> <span m="1969780">So</span> <span m="1969950">it's</span> <span m="1970060">a</span> <span m="1970100">lot</span> <span m="1970350">better</span> <span m="1970540">bound.</span> <span m="1971580">It's</span> <span m="1971700">1%</span> <span m="1972340">instead</span> <span m="1972580">of</span> <span m="1972660">40%.</span> <span m="1975150">So</span> <span m="1975250">knowing</span> <span m="1975660">the</span> <span m="1975730">variance</span> <span m="1976240">of</span> <span m="1976300">the</span> <span m="1976360">standard</span> <span m="1976680">deviation</span> <span m="1978540">gives</span> <span m="1978840">you</span> <span m="1978940">a</span> <span m="1979000">lot</span> <span m="1979290">more</span> <span m="1979460">information</span> <span m="1980020">and</span> <span m="1980150">generally</span> <span m="1980550">gives</span> <span m="1980710">you</span> <span m="1980800">much</span> <span m="1981090">better</span> <span m="1981280">bounds</span> <span m="1981870">on</span> <span m="1981940">the</span> <span m="1982000">probability</span> <span m="1982580">of</span> <span m="1982810">deviating</span> <span m="1984370">from</span> <span m="1985000">the</span> <span m="1985120">mean.</span> </p>
<p><span m="1986380">And</span> <span m="1986480">the</span> <span m="1986580">reason</span> <span m="1986790">it</span> <span m="1986850">gives</span> <span m="1987010">you</span> <span m="1987090">better</span> <span m="1987420">bounds</span> <span m="1987980">is</span> <span m="1988080">because</span> <span m="1988930">the</span> <span m="1989030">variance</span> <span m="1989850">is</span> <span m="1990000">squaring</span> <span m="1990820">deviations.</span> <span m="1992250">So</span> <span m="1992420">they</span> <span m="1992510">count</span> <span m="1992810">a</span> <span m="1992880">lot</span> <span m="1993130">more.</span> <span m="1996910">All</span> <span m="1997140">right,</span> <span m="1997650">now</span> <span m="2000250">let's</span> <span m="2000470">look</span> <span m="2000610">at</span> <span m="2000710">this</span> <span m="2000880">step</span> <span m="2001220">a</span> <span m="2001260">little</span> <span m="2001470">bit</span> <span m="2001640">more.</span> <span m="2016491">All</span> <span m="2016990">right,</span> <span m="2017220">let's</span> <span m="2019010">say</span> <span m="2019220">here</span> <span m="2019380">is</span> <span m="2019460">a</span> <span m="2019520">line,</span> <span m="2019990">and</span> <span m="2020100">here's</span> <span m="2020380">the</span> <span m="2020490">expected</span> <span m="2021040">value</span> <span m="2021480">of</span> <span m="2021590">R.</span> <span m="2023680">And</span> <span m="2024010">say</span> <span m="2024370">here's</span> <span m="2025510">10</span> <span m="2025830">standard</span> <span m="2026180">deviations</span> <span m="2028950">high</span> <span m="2029400">here.</span> <span m="2030340">So</span> <span m="2031140">this</span> <span m="2031320">will</span> <span m="2031410">be</span> <span m="2031490">more</span> <span m="2031810">than</span> <span m="2031950">10</span> <span m="2032180">standard</span> <span m="2032480">deviations.</span> <span m="2034090">And</span> <span m="2034440">this</span> <span m="2034570">will</span> <span m="2034660">be</span> <span m="2034750">10</span> <span m="2035020">standard</span> <span m="2035390">deviations</span> <span m="2035990">on</span> <span m="2036070">the</span> <span m="2036150">low</span> <span m="2036440">side.</span> <span m="2037850">So</span> <span m="2038000">here,</span> <span m="2038250">I'm</span> <span m="2038390">low.</span> </p>
<p><span m="2040710">Now,</span> <span m="2042690">this</span> <span m="2043020">line</span> <span m="2043390">here</span> <span m="2043670">with</span> <span m="2043760">the</span> <span m="2043870">absolute</span> <span m="2044280">value</span> <span m="2045580">is</span> <span m="2046200">figuring</span> <span m="2046530">out</span> <span m="2046640">the</span> <span m="2046710">probability</span> <span m="2047280">of</span> <span m="2047380">being</span> <span m="2047730">low</span> <span m="2049280">or</span> <span m="2049530">high.</span> <span m="2050300">This</span> <span m="2050610">is</span> <span m="2051190">the</span> <span m="2051429">probability</span> <span m="2051989">that</span> <span m="2052100">the</span> <span m="2052219">absolute</span> <span m="2052659">value</span> <span m="2053929">of</span> <span m="2054179">R</span> <span m="2054510">minus</span> <span m="2054860">its</span> <span m="2055020">expected</span> <span m="2055460">value</span> <span m="2058580">is</span> <span m="2058710">at</span> <span m="2058810">least</span> <span m="2059060">10</span> <span m="2059300">standard</span> <span m="2059600">deviations.</span> <span m="2063760">What</span> <span m="2063889">we</span> <span m="2064020">really</span> <span m="2064340">wanted</span> <span m="2064560">to</span> <span m="2064710">know</span> <span m="2065900">for</span> <span m="2066060">bound</span> <span m="2067280">was</span> <span m="2067469">just</span> <span m="2067750">the</span> <span m="2067840">high</span> <span m="2068090">side.</span> </p>
<p><span m="2072699">Now,</span> <span m="2073710">is</span> <span m="2073909">it</span> <span m="2074040">true</span> <span m="2074560">that</span> <span m="2074760">then,</span> <span m="2075190">since</span> <span m="2075469">the</span> <span m="2075570">probability</span> <span m="2076000">of</span> <span m="2076100">high</span> <span m="2076360">or</span> <span m="2076500">low</span> <span m="2076850">is</span> <span m="2076989">1</span> <span m="2077170">in</span> <span m="2077230">100,</span> <span m="2079110">the</span> <span m="2079179">probability</span> <span m="2079464">of</span> <span m="2079750">being</span> <span m="2080000">high</span> <span m="2080440">is</span> <span m="2080560">at</span> <span m="2080650">most</span> <span m="2080940">1</span> <span m="2081100">in</span> <span m="2081190">200,</span> <span m="2082570">half?</span> <span m="2085830">Is</span> <span m="2085980">that</span> <span m="2086130">true?</span> <span m="2088239">Yeah?</span> </p>
<p><span m="2089703">AUDIENCE: [INAUDIBLE]</span> </p>
<p></p>
<p><span m="2096610">PROFESSOR: Yeah,</span> <span m="2097350">it</span> <span m="2097520">is</span> <span m="2097670">not</span> <span m="2097940">necessarily</span> <span m="2098550">true</span> <span m="2098910">that</span> <span m="2099630">the</span> <span m="2099750">high</span> <span m="2100260">and</span> <span m="2100360">the</span> <span m="2100430">low</span> <span m="2100710">are</span> <span m="2100990">equal,</span> <span m="2101420">and</span> <span m="2101530">therefore</span> <span m="2101890">the</span> <span m="2101980">high</span> <span m="2102340">is</span> <span m="2102490">half</span> <span m="2103040">the</span> <span m="2103160">total.</span> <span m="2103625">It</span> <span m="2104090">might</span> <span m="2104410">be</span> <span m="2104520">true,</span> <span m="2105830">but</span> <span m="2105890">not</span> <span m="2106110">necessarily</span> <span m="2106680">true.</span> <span m="2106860">And</span> <span m="2106950">that's</span> <span m="2107070">a</span> <span m="2107110">mistake</span> <span m="2107480">that</span> <span m="2107850">often</span> <span m="2108230">gets</span> <span m="2108440">made</span> <span m="2109530">where</span> <span m="2109890">you'll</span> <span m="2110740">take</span> <span m="2113130">this</span> <span m="2113450">fact</span> <span m="2114050">as</span> <span m="2114220">being</span> <span m="2114430">less</span> <span m="2114650">than</span> <span m="2114750">100</span> <span m="2115180">to</span> <span m="2115250">conclude</span> <span m="2115710">that's</span> <span m="2116070">less</span> <span m="2116270">than</span> <span m="2116370">1</span> <span m="2116530">in</span> <span m="2116620">200.</span> <span m="2117090">And</span> <span m="2117200">that</span> <span m="2117460">you</span> <span m="2117560">can't</span> <span m="2117840">do,</span> <span m="2118820">unless</span> <span m="2119280">the</span> <span m="2119350">distribution</span> <span m="2119950">is</span> <span m="2120040">symmetric</span> <span m="2121730">around</span> <span m="2122130">the</span> <span m="2122200">expected</span> <span m="2122610">value.</span> <span m="2122950">Then</span> <span m="2123170">you</span> <span m="2123260">could</span> <span m="2123400">do</span> <span m="2123570">it,</span> <span m="2123650">if</span> <span m="2123770">it's</span> <span m="2123900">a</span> <span m="2123950">symmetric</span> <span m="2124490">distribution</span> <span m="2125600">around</span> <span m="2125940">the</span> <span m="2126000">expected</span> <span m="2126390">value.</span> <span m="2126790">But</span> <span m="2126990">usually</span> <span m="2127420">it's</span> <span m="2127600">not.</span> </p>
<p><span m="2131910">Now,</span> <span m="2132110">there</span> <span m="2132240">is</span> <span m="2132660">something</span> <span m="2133090">better</span> <span m="2133340">you</span> <span m="2133470">can</span> <span m="2133610">say.</span> <span m="2135660">So</span> <span m="2135860">let</span> <span m="2135980">me</span> <span m="2136250">tell</span> <span m="2136430">you</span> <span m="2136540">what</span> <span m="2136690">it</span> <span m="2136780">is.</span> <span m="2137100">But</span> <span m="2137230">we</span> <span m="2137320">won't</span> <span m="2137580">prove</span> <span m="2137860">it.</span> <span m="2139320">I</span> <span m="2139350">think</span> <span m="2139530">we</span> <span m="2139610">might</span> <span m="2139910">prove it</span> <span m="2140210">in</span> <span m="2140270">the</span> <span m="2140330">text.</span> <span m="2140720">I'm</span> <span m="2140800">not</span> <span m="2140980">sure.</span> <span m="2143200">If</span> <span m="2143350">you</span> <span m="2143460">just</span> <span m="2143890">want</span> <span m="2144100">the</span> <span m="2144190">high</span> <span m="2144490">side</span> <span m="2144850">or</span> <span m="2144890">just</span> <span m="2145130">want</span> <span m="2145280">the</span> <span m="2145350">low</span> <span m="2145590">side,</span> <span m="2145960">you</span> <span m="2146040">can</span> <span m="2146180">do</span> <span m="2146310">slightly</span> <span m="2147180">better</span> <span m="2148880">than</span> <span m="2149550">1</span> <span m="2149830">in</span> <span m="2149930">c</span> <span m="2150130">squared.</span> <span m="2154290">That's</span> <span m="2154540">the</span> <span m="2154600">following</span> <span m="2155740">theorem.</span> </p>
<p><span m="2159560">For</span> <span m="2159780">any</span> <span m="2161270">random</span> <span m="2161700">variable</span> <span m="2162700">R,</span> <span m="2165220">the</span> <span m="2165700">probability</span> <span m="2167320">that</span> <span m="2167900">R</span> <span m="2169980">is</span> <span m="2170170">on</span> <span m="2170300">the</span> <span m="2170410">high</span> <span m="2170730">side</span> <span m="2172500">by</span> <span m="2172980">c</span> <span m="2173390">standard</span> <span m="2173770">deviations</span> <span m="2176190">is</span> <span m="2176390">at</span> <span m="2176530">most</span> <span m="2176795">1</span> <span m="2177060">over</span> <span m="2177550">c</span> <span m="2178320">squared</span> <span m="2178850">plus</span> <span m="2179220">1.</span> <span m="2181030">So</span> <span m="2181180">it's</span> <span m="2181270">not</span> <span m="2181650">1</span> <span m="2181920">over</span> <span m="2182120">2c</span> <span m="2182370">squared.</span> <span m="2182640">It's</span> <span m="2183010">1</span> <span m="2183140">over c</span> <span m="2183470">squared</span> <span m="2183690">plus</span> <span m="2183920">1,</span> <span m="2184110">and</span> <span m="2184180">the</span> <span m="2184240">same</span> <span m="2184520">thing</span> <span m="2184710">for</span> <span m="2185430">the</span> <span m="2185540">probability</span> <span m="2186080">of</span> <span m="2186150">being</span> <span m="2186310">on</span> <span m="2186400">the</span> <span m="2186470">low</span> <span m="2186660">side.</span> </p>
<p><span m="2192300">Let's</span> <span m="2192670">see,</span> <span m="2194294">have</span> <span m="2194700">I</span> <span m="2194980">written</span> <span m="2195260">this</span> <span m="2195490">right?</span> <span m="2197474">Hmm,</span> <span m="2198460">I</span> <span m="2198530">want</span> <span m="2198710">to</span> <span m="2198770">get</span> <span m="2198890">this</span> <span m="2199110">as</span> <span m="2200710">less</span> <span m="2201340">than</span> <span m="2201530">or</span> <span m="2201660">equal</span> <span m="2201930">to</span> <span m="2202400">negative</span> <span m="2204050">c</span> <span m="2205230">times</span> <span m="2206470">the</span> <span m="2206530">standard</span> <span m="2206820">deviation.</span> <span m="2207540">So</span> <span m="2207640">here</span> <span m="2208010">I'm</span> <span m="2208560">high</span> <span m="2209040">by</span> <span m="2209220">c</span> <span m="2209590">or</span> <span m="2209640">more</span> <span m="2209830">standard</span> <span m="2210090">deviations.</span> <span m="2210940">Here</span> <span m="2211240">I'm</span> <span m="2211400">low.</span> <span m="2212040">So</span> <span m="2212280">R</span> <span m="2212580">is</span> <span m="2212730">less</span> <span m="2213200">than</span> <span m="2213330">the</span> <span m="2213410">expected</span> <span m="2213850">value</span> <span m="2214540">by</span> <span m="2214710">at</span> <span m="2214800">least</span> <span m="2215240">c</span> <span m="2215530">standard</span> <span m="2215940">deviations.</span> <span m="2217070">And</span> <span m="2217180">that</span> <span m="2217340">is</span> <span m="2217460">also</span> <span m="2217830">1</span> <span m="2218060">over c</span> <span m="2218470">squared</span> <span m="2218740">plus</span> <span m="2218980">1.</span> </p>
<p><span m="2220980">And</span> <span m="2221170">it</span> <span m="2221240">is</span> <span m="2221350">possible</span> <span m="2221950">to</span> <span m="2222030">find</span> <span m="2222690">distributions</span> <span m="2224740">that</span> <span m="2225390">hit</span> <span m="2225670">these</span> <span m="2225860">targets--</span> <span m="2226850">not</span> <span m="2227080">both</span> <span m="2227320">at</span> <span m="2227390">the</span> <span m="2227450">same</span> <span m="2227700">time,</span> <span m="2228040">but</span> <span m="2228150">one</span> <span m="2228630">or</span> <span m="2228700">the</span> <span m="2228830">other,</span> <span m="2229560">hit</span> <span m="2229710">those</span> <span m="2229900">targets.</span> <span m="2230310">So</span> <span m="2230440">that's</span> <span m="2230610">the</span> <span m="2230690">best</span> <span m="2231060">you</span> <span m="2231150">can</span> <span m="2231300">say</span> <span m="2231900">in</span> <span m="2232050">general.</span> <span m="2235120">All</span> <span m="2235320">right,</span> <span m="2235550">so</span> <span m="2235670">using</span> <span m="2236120">this</span> <span m="2236340">bound,</span> <span m="2236950">what's</span> <span m="2237300">the</span> <span m="2237390">probability</span> <span m="2238120">that</span> <span m="2238220">a</span> <span m="2238300">random</span> <span m="2238630">person</span> <span m="2238980">has</span> <span m="2239160">an</span> <span m="2239230">IQ</span> <span m="2239530">of</span> <span m="2239620">at</span> <span m="2239730">least</span> <span m="2239980">250?</span> <span m="2241931">It's</span> <span m="2242370">a</span> <span m="2242450">little</span> <span m="2242680">better</span> <span m="2242980">than</span> <span m="2243110">1</span> <span m="2243280">in</span> <span m="2243340">100.</span> </p>
<p><span m="2246106">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="2247070">PROFESSOR: Yeah,</span> <span m="2247380">1/101.</span> <span m="2248540">So</span> <span m="2248950">in</span> <span m="2249360">fact,</span> <span m="2249780">the</span> <span m="2249860">best</span> <span m="2250180">we</span> <span m="2250260">can</span> <span m="2250400">say</span> <span m="2250760">without</span> <span m="2251110">knowing</span> <span m="2251490">any</span> <span m="2251700">more</span> <span m="2251920">information</span> <span m="2252510">about</span> <span m="2252730">IQs</span> <span m="2255480">is</span> <span m="2255840">that</span> <span m="2255960">it's</span> <span m="2256670">at</span> <span m="2256810">most</span> <span m="2257190">1/101,</span> <span m="2259000">slightly</span> <span m="2259350">better.</span> <span m="2263610">Now</span> <span m="2263820">in</span> <span m="2263880">fact,</span> <span m="2264270">with</span> <span m="2264380">IQs,</span> <span m="2264870">they</span> <span m="2265000">know</span> <span m="2265230">more</span> <span m="2265480">about</span> <span m="2265690">the</span> <span m="2265760">distribution.</span> <span m="2267030">And</span> <span m="2267160">the</span> <span m="2267220">probability</span> <span m="2267780">is</span> <span m="2267880">a</span> <span m="2267960">lot</span> <span m="2268550">less.</span> <span m="2269800">Because</span> <span m="2270130">you know</span> <span m="2270395">more</span> <span m="2270660">about</span> <span m="2270840">the</span> <span m="2270910">distribution</span> <span m="2271470">than</span> <span m="2271600">we've</span> <span m="2271800">assumed</span> <span m="2272150">here.</span> <span m="2273670">In fact,</span> <span m="2273900">I</span> <span m="2273970">don't</span> <span m="2274110">think</span> <span m="2274250">anybody</span> <span m="2274580">has</span> <span m="2274820">an</span> <span m="2274890">IQ</span> <span m="2275200">over</span> <span m="2275350">250</span> <span m="2276720">as</span> <span m="2276850">far</span> <span m="2277040">as</span> <span m="2277140">I</span> <span m="2277200">know.</span> <span m="2278890">Any</span> <span m="2279490">questions</span> <span m="2281990">about</span> <span m="2282280">this?</span> </p>
<p><span m="2284860">OK,</span> <span m="2286030">all</span> <span m="2286250">right,</span> <span m="2286610">say</span> <span m="2287100">we</span> <span m="2287690">give</span> <span m="2287870">the</span> <span m="2287970">exam.</span> <span m="2289570">What</span> <span m="2289900">fraction</span> <span m="2290490">of</span> <span m="2290570">the</span> <span m="2290640">class</span> <span m="2292360">can</span> <span m="2292540">score</span> <span m="2293090">more</span> <span m="2293630">than</span> <span m="2293780">two</span> <span m="2294010">standard</span> <span m="2294410">deviations,</span> <span m="2295490">get</span> <span m="2295710">two</span> <span m="2295930">standard</span> <span m="2296210">deviations</span> <span m="2296690">or</span> <span m="2296920">more,</span> <span m="2297530">away</span> <span m="2298020">from</span> <span m="2298210">the</span> <span m="2298340">average,</span> <span m="2299190">above</span> <span m="2299510">or</span> <span m="2299570">below?</span> <span m="2303610">Could</span> <span m="2303850">half</span> <span m="2304180">the</span> <span m="2304280">class</span> <span m="2304790">be</span> <span m="2305510">two</span> <span m="2305710">standard</span> <span m="2306040">deviations</span> <span m="2306630">off</span> <span m="2307250">the</span> <span m="2307340">mean?</span> <span m="2309080">No?</span> <span m="2309640">What's</span> <span m="2309860">the</span> <span m="2309940">biggest</span> <span m="2310470">fraction</span> <span m="2310900">that</span> <span m="2310980">that</span> <span m="2311406">could happen?</span> </p>
<p><span m="2315090">What</span> <span m="2315430">do</span> <span m="2315550">I</span> <span m="2316003">do?</span> <span m="2316910">What</span> <span m="2317190">fraction</span> <span m="2317650">of</span> <span m="2317710">the</span> <span m="2317790">class</span> <span m="2318380">can</span> <span m="2318540">be</span> <span m="2320310">two</span> <span m="2320570">standard</span> <span m="2320850">deviations</span> <span m="2321290">or</span> <span m="2321490">more</span> <span m="2322860">from</span> <span m="2323130">the</span> <span m="2323190">mean?</span> <span m="2327690">What</span> <span m="2327870">is</span> <span m="2327980">it?</span> </p>
<p><span m="2328245">AUDIENCE: 1/4.</span> </p>
<p><span m="2329030">PROFESSOR: 1/4,</span> <span m="2329790">because</span> <span m="2329960">c</span> <span m="2330190">is</span> <span m="2330320">2.</span> <span m="2331246">You</span> <span m="2331710">don't</span> <span m="2331850">even</span> <span m="2332000">know</span> <span m="2332080">what</span> <span m="2332180">the</span> <span m="2332250">mean</span> <span m="2332490">is.</span> <span m="2332970">You</span> <span m="2333070">don't</span> <span m="2333220">know</span> <span m="2333290">what</span> <span m="2333380">the</span> <span m="2333440">standard</span> <span m="2333720">deviation</span> <span m="2334170">is.</span> <span m="2334360">You</span> <span m="2334420">don't</span> <span m="2334590">need</span> <span m="2334800">to.</span> <span m="2335620">I</span> <span m="2335720">just</span> <span m="2335940">asked,</span> <span m="2337200">you're</span> <span m="2337410">two</span> <span m="2337610">standard</span> <span m="2337910">deviations</span> <span m="2338490">off</span> <span m="2338720">or</span> <span m="2338800">more.</span> <span m="2339640">At</span> <span m="2339800">most,</span> <span m="2340040">1/4.</span> <span m="2343230">How</span> <span m="2343440">many</span> <span m="2343630">could</span> <span m="2343760">be</span> <span m="2343850">two</span> <span m="2344060">standard</span> <span m="2344400">deviations</span> <span m="2345050">high</span> <span m="2345740">or</span> <span m="2345940">better</span> <span m="2347080">at</span> <span m="2347240">most?</span> <span m="2349080">1/5--</span> <span m="2350200">1</span> <span m="2350360">over</span> <span m="2350560">4</span> <span m="2350780">plus</span> <span m="2351220">1,</span> <span m="2352430">good.</span> <span m="2353090">OK,</span> <span m="2355080">this</span> <span m="2355300">holds</span> <span m="2355510">true</span> <span m="2355720">no</span> <span m="2355830">matter</span> <span m="2356540">what</span> <span m="2357030">the</span> <span m="2357100">distribution</span> <span m="2357620">of</span> <span m="2357680">test</span> <span m="2357900">scores</span> <span m="2358260">is.</span> <span m="2358620">Yeah?</span> </p>
<p><span m="2359320">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="2361120">PROFESSOR: Which</span> <span m="2361340">one?</span> <span m="2362190">This</span> <span m="2362400">one?</span> </p>
<p><span m="2362840">AUDIENCE: Yeah.</span> </p>
<p><span m="2363540">PROFESSOR: Oh,</span> <span m="2363800">that's</span> <span m="2363990">more</span> <span m="2364140">complicated.</span> <span m="2365650">That'll</span> <span m="2366040">take us</span> <span m="2366300">several</span> <span m="2366580">boards</span> <span m="2366900">to</span> <span m="2367000">do,</span> <span m="2367750">to</span> <span m="2367860">prove</span> <span m="2368090">that.</span> <span m="2368850">And</span> <span m="2368970">I</span> <span m="2369010">forget</span> <span m="2369380">if</span> <span m="2369470">we</span> <span m="2369540">put</span> <span m="2369730">it</span> <span m="2369790">in</span> <span m="2369850">the</span> <span m="2369910">text</span> <span m="2370230">or</span> <span m="2370290">not.</span> <span m="2371240">It</span> <span m="2371710">might</span> <span m="2371940">be</span> <span m="2372020">in</span> <span m="2372090">the</span> <span m="2372150">text,</span> <span m="2373200">to</span> <span m="2373310">prove</span> <span m="2373780">that.</span> <span m="2376750">Any</span> <span m="2376960">other</span> <span m="2377250">questions?</span> </p>
<p><span m="2380830">OK</span> <span m="2384950">so</span> <span m="2385440">Markov</span> <span m="2385740">and</span> <span m="2386010">Chebyshev</span> <span m="2386280">are</span> <span m="2386590">sometimes</span> <span m="2387190">close,</span> <span m="2387770">sometimes</span> <span m="2388420">not.</span> <span m="2392800">Now,</span> <span m="2393280">for</span> <span m="2393430">the</span> <span m="2393510">rest</span> <span m="2393720">of</span> <span m="2393840">today,</span> <span m="2394260">we're</span> <span m="2394360">going</span> <span m="2394480">to</span> <span m="2394540">talk</span> <span m="2394770">about</span> <span m="2394960">a</span> <span m="2394990">much</span> <span m="2395490">more</span> <span m="2395660">powerful</span> <span m="2396310">technique.</span> <span m="2398710">But</span> <span m="2398780">it</span> <span m="2398870">only</span> <span m="2399100">works</span> <span m="2399440">in</span> <span m="2399520">a</span> <span m="2399560">special</span> <span m="2399960">case.</span> <span m="2400460">Now,</span> <span m="2400540">the</span> <span m="2400620">good</span> <span m="2400820">news</span> <span m="2401010">is</span> <span m="2401120">this</span> <span m="2401200">special</span> <span m="2401570">case</span> <span m="2401870">happens</span> <span m="2402280">all</span> <span m="2402560">the</span> <span m="2402650">time</span> <span m="2403230">in</span> <span m="2403370">practice.</span> <span m="2404950">And</span> <span m="2405110">it's</span> <span m="2405230">the</span> <span m="2405320">case</span> <span m="2405820">when</span> <span m="2406110">you're</span> <span m="2406330">analyzing</span> <span m="2407540">a</span> <span m="2407680">random</span> <span m="2408070">variable</span> <span m="2408550">that</span> <span m="2408670">itself</span> <span m="2409450">is</span> <span m="2409650">the</span> <span m="2409750">sum</span> <span m="2410870">of</span> <span m="2411000">a</span> <span m="2411060">bunch</span> <span m="2411290">of</span> <span m="2411420">other</span> <span m="2411680">random</span> <span m="2411960">variables.</span> <span m="2412640">And</span> <span m="2412750">we've</span> <span m="2412870">seen</span> <span m="2413280">already</span> <span m="2413570">examples</span> <span m="2414180">like</span> <span m="2414500">that.</span> </p>
<p><span m="2415600">And</span> <span m="2416230">the</span> <span m="2416350">other</span> <span m="2416550">random</span> <span m="2416810">variables</span> <span m="2417190">have</span> <span m="2417520">to</span> <span m="2417630">be</span> <span m="2418050">mutually</span> <span m="2418450">independent.</span> <span m="2420100">And</span> <span m="2420180">in</span> <span m="2420260">this</span> <span m="2420440">case,</span> <span m="2420870">you</span> <span m="2420970">get</span> <span m="2421010">a</span> <span m="2421410">bound</span> <span m="2421730">that's</span> <span m="2421870">called</span> <span m="2422240">a</span> <span m="2422340">Chernoff</span> <span m="2422840">bound.</span> <span m="2423900">And</span> <span m="2424080">this</span> <span m="2424240">is</span> <span m="2424340">the</span> <span m="2424430">same</span> <span m="2424770">Chernoff</span> <span m="2425720">who</span> <span m="2425860">figured</span> <span m="2426150">out</span> <span m="2426280">how</span> <span m="2426390">to</span> <span m="2426480">beat</span> <span m="2426690">the</span> <span m="2426770">lottery.</span> <span m="2428390">And</span> <span m="2429320">it's</span> <span m="2429430">interesting.</span> <span m="2431370">Long</span> <span m="2431670">after</span> <span m="2431880">we</span> <span m="2431970">started</span> <span m="2432360">teaching</span> <span m="2432720">this,</span> <span m="2433420">originally</span> <span m="2433770">this</span> <span m="2433900">stuff</span> <span m="2434100">was</span> <span m="2434230">only</span> <span m="2434390">taught,</span> <span m="2434770">for</span> <span m="2434890">Chernoff</span> <span m="2435230">bounds,</span> <span m="2435820">for</span> <span m="2435930">graduate</span> <span m="2436360">students.</span> <span m="2437280">And</span> <span m="2437410">now</span> <span m="2437610">we</span> <span m="2438030">teach</span> <span m="2438230">it</span> <span m="2438300">here.</span> <span m="2438420">Because</span> <span m="2438580">it's</span> <span m="2438720">so</span> <span m="2438900">important.</span> <span m="2439155">And</span> <span m="2439410">it</span> <span m="2439500">really</span> <span m="2439830">is</span> <span m="2439970">accessible.</span> </p>
<p><span m="2441560">It'll</span> <span m="2441680">be</span> <span m="2441810">probably</span> <span m="2442680">the</span> <span m="2442740">most</span> <span m="2442910">complicated</span> <span m="2443430">proof</span> <span m="2443700">we've</span> <span m="2443840">done</span> <span m="2444250">to</span> <span m="2444815">establish</span> <span m="2445150">a</span> <span m="2445200">Chernoff</span> <span m="2445380">bound.</span> <span m="2446330">But</span> <span m="2446520">Chernoff</span> <span m="2446870">himself,</span> <span m="2447900">when</span> <span m="2448160">he</span> <span m="2448230">discovered</span> <span m="2448730">this,</span> <span m="2449400">thought</span> <span m="2449580">it</span> <span m="2449640">was</span> <span m="2449750">no</span> <span m="2449900">big</span> <span m="2450090">deal.</span> <span m="2450840">In fact,</span> <span m="2451010">he</span> <span m="2451090">couldn't</span> <span m="2451350">figure</span> <span m="2451650">out</span> <span m="2451820">why</span> <span m="2451970">everybody in</span> <span m="2452340">computer</span> <span m="2452660">science</span> <span m="2453030">was</span> <span m="2453150">always</span> <span m="2453500">writing</span> <span m="2453750">papers</span> <span m="2454120">with</span> <span m="2454270">Chernoff</span> <span m="2454650">bounds in</span> <span m="2455100">them.</span> </p>
<p><span m="2456290">And</span> <span m="2456380">that's</span> <span m="2456540">because</span> <span m="2457370">he</span> <span m="2457570">didn't</span> <span m="2457730">put</span> <span m="2457870">any</span> <span m="2457970">emphasis</span> <span m="2458460">on</span> <span m="2458560">the</span> <span m="2458620">bounds</span> <span m="2458930">in</span> <span m="2459000">his</span> <span m="2459220">work.</span> <span m="2459760">But</span> <span m="2459930">computer</span> <span m="2460250">scientists</span> <span m="2460690">who</span> <span m="2460770">came</span> <span m="2461050">later</span> <span m="2461800">found</span> <span m="2462050">all</span> <span m="2462260">sorts</span> <span m="2462540">of</span> <span m="2462610">important</span> <span m="2462980">applications.</span> <span m="2464180">And</span> <span m="2464320">we'll</span> <span m="2464430">see</span> <span m="2464610">some</span> <span m="2464840">of</span> <span m="2464900">those</span> <span m="2465830">today.</span> <span m="2466920">So</span> <span m="2467070">let</span> <span m="2467200">me</span> <span m="2467290">tell</span> <span m="2467520">you</span> <span m="2468040">what</span> <span m="2468260">the</span> <span m="2468330">bound</span> <span m="2468740">is.</span> <span m="2471060">And</span> <span m="2471180">the</span> <span m="2471280">nice</span> <span m="2471370">thing</span> <span m="2471590">is</span> <span m="2471870">it</span> <span m="2472010">really</span> <span m="2472270">is</span> <span m="2472370">Markov's</span> <span m="2472680">theorem</span> <span m="2473090">again</span> <span m="2473340">in</span> <span m="2473440">disguise,</span> <span m="2474940">just</span> <span m="2475270">a</span> <span m="2475300">little</span> <span m="2475460">more</span> <span m="2475580">complicated.</span> </p>
<p><span m="2491230">Theorem--</span> <span m="2492366">it's</span> <span m="2492730">called</span> <span m="2493010">a</span> <span m="2493100">Chernoff</span> <span m="2493530">bound.</span> <span m="2502470">Let</span> <span m="2503710">T1,</span> <span m="2504810">T2,</span> <span m="2506740">up</span> <span m="2506930">to</span> <span m="2507010">Tn</span> <span m="2509550">be</span> <span m="2510000">any</span> <span m="2511920">mutually</span> <span m="2512590">independent--</span> <span m="2515620">that's</span> <span m="2515930">really</span> <span m="2516190">important--</span> <span m="2518940">random</span> <span m="2519290">variables</span> <span m="2521090">such</span> <span m="2521520">that</span> <span m="2526380">each</span> <span m="2526670">of</span> <span m="2526790">them</span> <span m="2527920">takes</span> <span m="2528200">values</span> <span m="2528780">only</span> <span m="2529130">between</span> <span m="2529470">0</span> <span m="2529740">and</span> <span m="2529860">1.</span> <span m="2532110">And</span> <span m="2532270">if</span> <span m="2532340">they</span> <span m="2532460">don't,</span> <span m="2533420">just</span> <span m="2533690">normalize</span> <span m="2534240">them</span> <span m="2534350">so</span> <span m="2534500">they</span> <span m="2534610">do.</span> <span m="2536500">So</span> <span m="2536620">we're</span> <span m="2536700">going</span> <span m="2536820">to</span> <span m="2536890">take</span> <span m="2537220">a</span> <span m="2537280">bunch</span> <span m="2537550">of</span> <span m="2537630">random</span> <span m="2537890">variables</span> <span m="2538290">that</span> <span m="2538380">are</span> <span m="2538420">mutually</span> <span m="2538790">independent.</span> <span m="2539340">And</span> <span m="2539460">they</span> <span m="2540430">are</span> <span m="2540560">all</span> <span m="2540720">between</span> <span m="2541080">0</span> <span m="2541400">and</span> <span m="2541540">1.</span> </p>
<p><span m="2546120">Then</span> <span m="2546390">we're</span> <span m="2546480">going</span> <span m="2546610">to</span> <span m="2546670">look</span> <span m="2546830">at</span> <span m="2546900">the</span> <span m="2546970">sum</span> <span m="2552010">of</span> <span m="2552150">those</span> <span m="2552390">random</span> <span m="2552660">variables,</span> <span m="2556060">call</span> <span m="2556330">that</span> <span m="2556570">T.</span> <span m="2560630">Then</span> <span m="2562230">for</span> <span m="2562440">any</span> <span m="2562730">c</span> <span m="2564880">at</span> <span m="2565040">least</span> <span m="2565340">1,</span> <span m="2568250">the</span> <span m="2568350">probability</span> <span m="2569700">that</span> <span m="2569870">the</span> <span m="2569970">sum</span> <span m="2570510">random</span> <span m="2570800">variable</span> <span m="2571270">is</span> <span m="2571450">at</span> <span m="2571560">least</span> <span m="2571990">c</span> <span m="2572420">times</span> <span m="2573000">its</span> <span m="2573140">expected</span> <span m="2573650">value.</span> <span m="2576060">So</span> <span m="2576320">it's</span> <span m="2576480">going to be</span> <span m="2576620">the</span> <span m="2576720">high</span> <span m="2576940">side</span> <span m="2577230">here--</span> <span m="2578000">is</span> <span m="2578230">at</span> <span m="2578650">most</span> <span m="2579590">e</span> <span m="2579910">to</span> <span m="2579990">the</span> <span m="2580070">minus</span> <span m="2580495">z,</span> <span m="2581320">and I'll</span> <span m="2581920">tell</span> <span m="2582070">you</span> <span m="2582130">what</span> <span m="2582240">that</span> <span m="2582400">is</span> <span m="2582530">in</span> <span m="2582610">a</span> <span m="2582660">minute,</span> <span m="2583600">times</span> <span m="2583930">the</span> <span m="2584000">expected</span> <span m="2584450">value</span> <span m="2584760">of</span> <span m="2585252">T</span> <span m="2587220">where</span> <span m="2587800">z</span> <span m="2593360">is</span> <span m="2595380">c</span> <span m="2596330">natural</span> <span m="2596770">log</span> <span m="2597080">of</span> <span m="2597180">c</span> <span m="2599130">plus</span> <span m="2599470">1</span> <span m="2600170">minus</span> <span m="2600510">c.</span> <span m="2601280">And</span> <span m="2601430">it</span> <span m="2601500">turns</span> <span m="2601860">out</span> <span m="2602540">if</span> <span m="2602690">c</span> <span m="2602900">is</span> <span m="2603030">bigger</span> <span m="2603240">than</span> <span m="2603390">1,</span> <span m="2603830">this</span> <span m="2604060">is</span> <span m="2604160">positive.</span> </p>
<p><span m="2609300">So</span> <span m="2609390">that's</span> <span m="2609590">a</span> <span m="2609640">lot,</span> <span m="2610250">one</span> <span m="2611270">of</span> <span m="2611330">the</span> <span m="2611390">longest</span> <span m="2611880">theorems</span> <span m="2612080">we</span> <span m="2612160">wrote</span> <span m="2612380">down</span> <span m="2612620">here.</span> <span m="2614730">But</span> <span m="2614940">what</span> <span m="2615040">it</span> <span m="2615140">says</span> <span m="2615500">is</span> <span m="2615650">that</span> <span m="2615880">probability</span> <span m="2616370">were</span> <span m="2616520">high</span> <span m="2618910">is</span> <span m="2619310">exponentially</span> <span m="2620190">small.</span> <span m="2622610">As</span> <span m="2622770">the</span> <span m="2622860">expected</span> <span m="2623340">value</span> <span m="2623750">is</span> <span m="2623900">big,</span> <span m="2624230">the</span> <span m="2624290">chance</span> <span m="2624650">of</span> <span m="2624730">being</span> <span m="2625050">high</span> <span m="2625460">gets</span> <span m="2625730">really,</span> <span m="2626180">really</span> <span m="2626320">tiny.</span> <span m="2628340">Now,</span> <span m="2628820">I'm</span> <span m="2629000">going</span> <span m="2629120">to</span> <span m="2629180">prove</span> <span m="2629490">it</span> <span m="2629570">in</span> <span m="2629630">a</span> <span m="2629680">minute.</span> <span m="2630100">But</span> <span m="2630970">let's</span> <span m="2631160">just</span> <span m="2631420">plug in</span> <span m="2631770">some</span> <span m="2631920">examples</span> <span m="2632560">to</span> <span m="2633100">see</span> <span m="2633330">what's</span> <span m="2633510">going</span> <span m="2633740">on</span> <span m="2633900">here.</span> </p>
<p><span m="2645200">So</span> <span m="2647600">for</span> <span m="2647680">example,</span> <span m="2650730">suppose</span> <span m="2653780">the</span> <span m="2653930">expected</span> <span m="2654490">value</span> <span m="2654810">of</span> <span m="2654890">T</span> <span m="2657270">is</span> <span m="2657470">100.</span> <span m="2660080">And</span> <span m="2660450">suppose</span> <span m="2661990">c</span> <span m="2662590">is</span> <span m="2663550">2.</span> <span m="2665230">So</span> <span m="2665420">we</span> <span m="2665520">expect</span> <span m="2665900">to</span> <span m="2665990">have</span> <span m="2666130">100</span> <span m="2667520">come</span> <span m="2667740">out</span> <span m="2667870">of</span> <span m="2667930">the</span> <span m="2668000">sum.</span> <span m="2668990">The</span> <span m="2669090">probability</span> <span m="2669740">we</span> <span m="2669890">get</span> <span m="2670240">at</span> <span m="2670400">least</span> <span m="2670840">200--</span> <span m="2672810">well,</span> <span m="2673940">let's</span> <span m="2674160">figure</span> <span m="2674410">out</span> <span m="2674520">what</span> <span m="2674640">that</span> <span m="2674820">is.</span> <span m="2675940">c</span> <span m="2676270">being</span> <span m="2676530">2</span> <span m="2676800">we</span> <span m="2676910">can</span> <span m="2677030">evaluate</span> <span m="2677520">z</span> <span m="2677860">now.</span> <span m="2679450">It's</span> <span m="2679650">2</span> <span m="2680070">natural</span> <span m="2680550">log</span> <span m="2680810">of</span> <span m="2680930">2</span> <span m="2681940">plus</span> <span m="2682300">1</span> <span m="2682660">minus</span> <span m="2683010">2.</span> <span m="2683990">And</span> <span m="2684100">that's</span> <span m="2684280">close</span> <span m="2684680">to</span> <span m="2684920">but</span> <span m="2685020">a</span> <span m="2685060">little</span> <span m="2685280">larger</span> <span m="2685690">than</span> <span m="2686030">0.38.</span> </p>
<p><span m="2688990">So</span> <span m="2689140">we</span> <span m="2689210">can</span> <span m="2689340">plug</span> <span m="2689795">z</span> <span m="2690250">in,</span> <span m="2690820">the</span> <span m="2690950">exponent</span> <span m="2691360">up</span> <span m="2691480">there,</span> <span m="2691720">and</span> <span m="2691830">find</span> <span m="2692180">that the</span> <span m="2692250">probability</span> <span m="2692930">that</span> <span m="2693080">T</span> <span m="2693430">is</span> <span m="2693560">at</span> <span m="2693690">least</span> <span m="2693960">twice</span> <span m="2694330">its</span> <span m="2694460">expected</span> <span m="2694910">value,</span> <span m="2695440">namely</span> <span m="2696450">at least</span> <span m="2696700">200,</span> <span m="2698340">is</span> <span m="2698490">at</span> <span m="2698610">most</span> <span m="2699420">e to</span> <span m="2699750">the</span> <span m="2699820">minus</span> <span m="2700390">0.38</span> <span m="2702130">times</span> <span m="2702440">100,</span> <span m="2704560">which</span> <span m="2704720">is</span> <span m="2705380">e to</span> <span m="2705740">the</span> <span m="2705820">minus</span> <span m="2706340">38,</span> <span m="2709270">which</span> <span m="2709540">is</span> <span m="2710010">just</span> <span m="2710290">really</span> <span m="2710570">small.</span> </p>
<p><span m="2713620">So</span> <span m="2713700">that's</span> <span m="2713920">just</span> <span m="2714160">way</span> <span m="2714690">better</span> <span m="2714960">than</span> <span m="2715110">any</span> <span m="2715390">results</span> <span m="2715850">you</span> <span m="2715940">get</span> <span m="2716200">with</span> <span m="2716420">Markov</span> <span m="2717420">or</span> <span m="2717560">Chebyshev.</span> <span m="2720830">So</span> <span m="2720990">if</span> <span m="2721090">you</span> <span m="2721200">have</span> <span m="2721320">a</span> <span m="2721370">bunch</span> <span m="2721560">of</span> <span m="2721630">random</span> <span m="2721860">variables</span> <span m="2722250">between</span> <span m="2722480">0 and 1,</span> <span m="2723320">and</span> <span m="2723470">they're</span> <span m="2723560">mutually</span> <span m="2723750">independent,</span> <span m="2724570">you</span> <span m="2724730">add</span> <span m="2724920">them</span> <span m="2725040">up.</span> <span m="2725560">If</span> <span m="2725740">you</span> <span m="2725850">expect</span> <span m="2726260">100</span> <span m="2727010">as</span> <span m="2727180">the</span> <span m="2727290">answer,</span> <span m="2728350">the</span> <span m="2728450">chance</span> <span m="2728780">of</span> <span m="2728850">getting</span> <span m="2729070">200</span> <span m="2729450">or</span> <span m="2729490">more--</span> <span m="2729950">forget</span> <span m="2730270">about</span> <span m="2730540">it,</span> <span m="2730940">not</span> <span m="2731130">going</span> <span m="2731250">to</span> <span m="2731320">happen.</span> </p>
<p><span m="2735040">Now,</span> <span m="2735920">of</span> <span m="2735990">course</span> <span m="2736220">Chernoff</span> <span m="2736540">doesn't</span> <span m="2736780">apply</span> <span m="2737040">to</span> <span m="2737130">all</span> <span m="2737460">distributions.</span> <span m="2738150">It</span> <span m="2738240">has</span> <span m="2738490">to</span> <span m="2738560">be</span> <span m="2738720">this</span> <span m="2738890">type.</span> <span m="2740160">This</span> <span m="2740400">is</span> <span m="2740660">a</span> <span m="2740720">pretty</span> <span m="2741050">broad</span> <span m="2741640">class.</span> <span m="2742280">In</span> <span m="2742360">fact,</span> <span m="2743160">it</span> <span m="2743310">contains</span> <span m="2744240">the</span> <span m="2744320">class</span> <span m="2744680">of</span> <span m="2744780">all</span> <span m="2744930">Bernoulli</span> <span m="2745340">distributions.</span> <span m="2747200">So</span> <span m="2747340">I have</span> <span m="2747390">binomial</span> <span m="2747850">distributions.</span> <span m="2750130">Because</span> <span m="2750310">remember</span> <span m="2750580">a</span> <span m="2750630">binomial</span> <span m="2751200">distribution--</span> <span m="2753750">well,</span> <span m="2754620">remember</span> <span m="2754850">binomial</span> <span m="2755410">distributions?</span> <span m="2756090">That's</span> <span m="2756340">where</span> <span m="2756520">T</span> <span m="2756800">is</span> <span m="2756940">the</span> <span m="2757020">sum</span> <span m="2757260">of</span> <span m="2757370">Ti's.</span> </p>
<p><span m="2758830">In</span> <span m="2758970">binomial,</span> <span m="2760180">you</span> <span m="2760320">have</span> <span m="2760550">Tj</span> <span m="2761250">is</span> <span m="2761430">0</span> <span m="2761950">or</span> <span m="2762280">1.</span> <span m="2762690">It</span> <span m="2763160">can't</span> <span m="2763430">be</span> <span m="2763530">in</span> <span m="2763610">between.</span> <span m="2765210">And</span> <span m="2765240">with</span> <span m="2765350">binomial,</span> <span m="2766420">all</span> <span m="2766750">Tj's</span> <span m="2767380">have</span> <span m="2767560">the</span> <span m="2767640">same</span> <span m="2768260">distribution.</span> <span m="2769660">With</span> <span m="2769810">Chernoff,</span> <span m="2770410">they</span> <span m="2770550">can</span> <span m="2770670">all</span> <span m="2770800">be</span> <span m="2770900">different.</span> <span m="2772840">So</span> <span m="2773010">Chernoff</span> <span m="2773440">is</span> <span m="2773560">much</span> <span m="2774030">broader</span> <span m="2774430">than</span> <span m="2774580">binomial.</span> <span m="2775730">The</span> <span m="2775870">individual</span> <span m="2776420">guys</span> <span m="2776740">here</span> <span m="2776950">can</span> <span m="2777080">have</span> <span m="2777190">different</span> <span m="2777510">distributions</span> <span m="2778590">and</span> <span m="2778730">attain</span> <span m="2779120">values</span> <span m="2779550">anywhere</span> <span m="2779930">between</span> <span m="2780330">0</span> <span m="2780610">and</span> <span m="2780730">1,</span> <span m="2781640">as</span> <span m="2781830">opposed</span> <span m="2782300">to</span> <span m="2782480">just</span> <span m="2782780">one</span> <span m="2782940">or</span> <span m="2783010">the</span> <span m="2783140">other.</span> <span m="2786062">Any</span> <span m="2786550">questions</span> <span m="2787030">about</span> <span m="2788260">this</span> <span m="2789850">theorem</span> <span m="2790330">and</span> <span m="2790470">what</span> <span m="2790590">it</span> <span m="2791540">says</span> <span m="2792000">in</span> <span m="2794050">the</span> <span m="2794170">terms</span> <span m="2794490">there?</span> </p>
<p><span m="2797630">One</span> <span m="2798000">nice</span> <span m="2798220">thing</span> <span m="2798380">about</span> <span m="2798640">it</span> <span m="2798730">is</span> <span m="2799080">the</span> <span m="2799170">number</span> <span m="2799640">of</span> <span m="2799770">random</span> <span m="2800040">variables</span> <span m="2801350">doesn't</span> <span m="2801600">even</span> <span m="2801760">show</span> <span m="2801980">up</span> <span m="2802100">in</span> <span m="2802170">the</span> <span m="2802250">answer</span> <span m="2802570">here.</span> <span m="2804010">n</span> <span m="2804240">doesn't</span> <span m="2804440">even</span> <span m="2804580">appear.</span> <span m="2806310">Yeah.</span> </p>
<p><span m="2807190">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="2808950">PROFESSOR: Does</span> <span m="2809070">not</span> <span m="2809250">apply</span> <span m="2809330">to</span> <span m="2809520">what?</span> </p>
<p><span m="2810173">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="2812590">PROFESSOR: Yeah,</span> <span m="2812890">when</span> <span m="2812970">c</span> <span m="2813070">equals</span> <span m="2813610">1,</span> <span m="2814910">what</span> <span m="2815130">happens</span> <span m="2815620">is</span> <span m="2815870">z</span> <span m="2815990">is</span> <span m="2816100">0.</span> <span m="2817700">Because</span> <span m="2817870">I</span> <span m="2817940">have</span> <span m="2818060">a</span> <span m="2818100">log</span> <span m="2818430">of</span> <span m="2818530">1</span> <span m="2818770">is</span> <span m="2818870">0,</span> <span m="2819800">and</span> <span m="2819980">1</span> <span m="2820200">minus</span> <span m="2820600">1</span> <span m="2820810">is</span> <span m="2820940">0.</span> <span m="2821230">And</span> <span m="2821520">if</span> <span m="2821730">z</span> <span m="2822060">is</span> <span m="2822180">0,</span> <span m="2823320">it</span> <span m="2823470">says</span> <span m="2823770">your</span> <span m="2823880">probability</span> <span m="2824540">is</span> <span m="2824910">upper</span> <span m="2825160">bounded</span> <span m="2825550">by</span> <span m="2825820">1.</span> <span m="2827360">Well,</span> <span m="2828320">not</span> <span m="2828510">too</span> <span m="2828670">interesting,</span> <span m="2829070">because</span> <span m="2829300">any</span> <span m="2829540">probability</span> <span m="2830080">is</span> <span m="2830210">upper</span> <span m="2830460">bounded</span> <span m="2830790">by</span> <span m="2830920">1.</span> <span m="2832110">So</span> <span m="2832320">it</span> <span m="2832390">doesn't</span> <span m="2832700">give</span> <span m="2832950">you</span> <span m="2833110">any</span> <span m="2833380">information</span> <span m="2834070">when</span> <span m="2834200">c</span> <span m="2834400">is</span> <span m="2834520">0,</span> <span m="2835750">none</span> <span m="2835980">at</span> <span m="2836070">all.</span> <span m="2836590">But</span> <span m="2836750">as</span> <span m="2836830">soon</span> <span m="2837020">as</span> <span m="2837190">c</span> <span m="2837320">starts</span> <span m="2837640">being--</span> <span m="2838060">sorry, if</span> <span m="2838330">c</span> <span m="2838490">is</span> <span m="2838620">1.</span> <span m="2838970">As</span> <span m="2839210">soon</span> <span m="2839410">as</span> <span m="2839470">c</span> <span m="2839720">starts</span> <span m="2839960">being</span> <span m="2840190">bigger</span> <span m="2840380">than</span> <span m="2840530">1,</span> <span m="2841460">which</span> <span m="2841680">is</span> <span m="2841790">sort</span> <span m="2841980">of</span> <span m="2842050">a</span> <span m="2842140">case</span> <span m="2842480">you're</span> <span m="2842590">interested</span> <span m="2842950">in,</span> <span m="2843130">you're</span> <span m="2843830">bigger</span> <span m="2844090">than</span> <span m="2844210">your</span> <span m="2844410">expectation,</span> <span m="2845530">then</span> <span m="2845750">it</span> <span m="2845850">gives</span> <span m="2846200">very</span> <span m="2846700">powerful</span> <span m="2847280">results.</span> <span m="2848720">Yeah.</span> </p>
<p><span m="2849974">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="2853140">PROFESSOR: Yeah,</span> <span m="2853430">you</span> <span m="2853590">can.</span> <span m="2854480">It's</span> <span m="2854680">true</span> <span m="2854910">for</span> <span m="2855060">n</span> <span m="2855250">equals</span> <span m="2855480">1</span> <span m="2855690">as</span> <span m="2855820">well.</span> <span m="2856990">Now,</span> <span m="2858366">it</span> <span m="2858780">doesn't</span> <span m="2859140">give</span> <span m="2859290">you</span> <span m="2859380">a</span> <span m="2859450">lot</span> <span m="2859760">of</span> <span m="2859840">information.</span> <span m="2860450">Because</span> <span m="2861950">if</span> <span m="2862090">c</span> <span m="2862310">is</span> <span m="2862460">bigger</span> <span m="2862680">than</span> <span m="2862820">1</span> <span m="2864090">and</span> <span m="2864360">n was</span> <span m="2864520">1,</span> <span m="2865060">so</span> <span m="2865140">it's</span> <span m="2865210">using</span> <span m="2865430">one</span> <span m="2865630">variable,</span> <span m="2866200">what's</span> <span m="2866440">the</span> <span m="2866510">probability</span> <span m="2867060">that</span> <span m="2867180">a</span> <span m="2867250">random</span> <span m="2867570">variable</span> <span m="2867990">exceeds</span> <span m="2871940">its</span> <span m="2872150">expectation,</span> <span m="2873090">c</span> <span m="2873560">times</span> <span m="2873970">its</span> <span m="2874090">expectation?</span> </p>
<p><span m="2876468">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="2878850">PROFESSOR: Yeah,</span> <span m="2879190">let's</span> <span m="2879640">see</span> <span m="2880030">now.</span> <span m="2880790">Maybe</span> <span m="2880990">it</span> <span m="2881070">does</span> <span m="2881290">give</span> <span m="2881440">you</span> <span m="2881540">information.</span> <span m="2883410">Because</span> <span m="2883560">the</span> <span m="2883660">random</span> <span m="2883950">variable</span> <span m="2884250">has</span> <span m="2884400">a</span> <span m="2884440">distribution</span> <span m="2884970">on</span> <span m="2885090">0,</span> <span m="2885450">1.</span> <span m="2890060">That's</span> <span m="2890280">right,</span> <span m="2890380">so</span> <span m="2890480">it</span> <span m="2890540">does</span> <span m="2890770">give</span> <span m="2890910">you</span> <span m="2890990">some</span> <span m="2891140">information.</span> <span m="2891860">But</span> <span m="2891970">I</span> <span m="2892020">don't</span> <span m="2892310">think</span> <span m="2892490">it</span> <span m="2892580">gives</span> <span m="2892890">you</span> <span m="2894730">a</span> <span m="2894930">lot.</span> <span m="2895270">I</span> <span m="2895490">have</span> <span m="2895600">to</span> <span m="2895670">think</span> <span m="2895880">about</span> <span m="2896140">that.</span> <span m="2897280">What</span> <span m="2897510">happens</span> <span m="2897880">when</span> <span m="2897970">there's</span> <span m="2898090">just</span> <span m="2898300">one</span> <span m="2898480">guy?</span> </p>
<p><span m="2898620">Because</span> <span m="2898790">the</span> <span m="2898870">same</span> <span m="2899140">thing</span> <span m="2899320">is</span> <span m="2899420">true.</span> <span m="2899535">It's</span> <span m="2899650">just</span> <span m="2899850">now</span> <span m="2899990">for</span> <span m="2900140">a</span> <span m="2900200">single</span> <span m="2901410">random</span> <span m="2901770">variable</span> <span m="2902140">on</span> <span m="2902310">0,</span> <span m="2902720">1</span> <span m="2904180">the</span> <span m="2904260">chance</span> <span m="2904700">that</span> <span m="2904840">your</span> <span m="2907260">twice</span> <span m="2907580">the</span> <span m="2907690">expected</span> <span m="2908240">value.</span> <span m="2908960">I have</span> <span m="2909040">to</span> <span m="2909110">think</span> <span m="2909310">about</span> <span m="2909550">that.</span> <span m="2909800">That's</span> <span m="2909970">a</span> <span m="2910020">good</span> <span m="2910150">question.</span> <span m="2911320">Does</span> <span m="2911390">it</span> <span m="2911440">do</span> <span m="2911630">anything</span> <span m="2911870">interesting</span> <span m="2912300">there?</span> <span m="2914400">OK,</span> <span m="2917080">all</span> <span m="2917240">right,</span> <span m="2917470">so</span> <span m="2917600">let's</span> <span m="2918710">do</span> <span m="2918980">an</span> <span m="2919060">example</span> <span m="2920420">of</span> <span m="2920740">how</span> <span m="2920910">you</span> <span m="2921000">might apply</span> <span m="2921490">this.</span> </p>
<p><span m="2936400">Say</span> <span m="2936710">that</span> <span m="2936870">you're</span> <span m="2937510">playing</span> <span m="2937920">Pick</span> <span m="2938120">4,</span> <span m="2940210">and</span> <span m="2940670">10</span> <span m="2940950">million</span> <span m="2941250">people</span> <span m="2941560">are</span> <span m="2941640">playing.</span> <span m="2951140">And</span> <span m="2951440">say</span> <span m="2951720">in</span> <span m="2951800">this</span> <span m="2952010">version</span> <span m="2952390">of</span> <span m="2952490">Pick</span> <span m="2952720">4,</span> <span m="2954330">you're</span> <span m="2954490">picking</span> <span m="2954750">a</span> <span m="2954800">four</span> <span m="2955000">digit</span> <span m="2955270">number,</span> <span m="2956210">four</span> <span m="2956480">single</span> <span m="2956760">digits.</span> <span m="2957820">And</span> <span m="2958360">you</span> <span m="2958450">win</span> <span m="2958750">if</span> <span m="2958830">you</span> <span m="2958900">get</span> <span m="2959040">an</span> <span m="2959130">exact</span> <span m="2959560">match.</span> <span m="2961050">So</span> <span m="2961210">the</span> <span m="2961300">probability</span> <span m="2961880">of</span> <span m="2961970">winning,</span> <span m="2964130">a</span> <span m="2964230">person</span> <span m="2964630">winning,</span> <span m="2965820">well,</span> <span m="2966020">they've</span> <span m="2966110">got</span> <span m="2966280">to</span> <span m="2966340">get</span> <span m="2966480">all</span> <span m="2966620">four</span> <span m="2966860">digits</span> <span m="2967250">right.</span> <span m="2968170">That's</span> <span m="2968410">1</span> <span m="2968600">in</span> <span m="2968700">10,000,</span> <span m="2970180">10 to</span> <span m="2970250">the</span> <span m="2970330">fourth.</span> </p>
<p><span m="2974510">What's</span> <span m="2974770">the</span> <span m="2974890">expected</span> <span m="2975470">number</span> <span m="2975720">of</span> <span m="2975810">winners?</span> <span m="2980620">If</span> <span m="2981040">I</span> <span m="2981170">got</span> <span m="2981370">10</span> <span m="2981610">million</span> <span m="2981930">people,</span> <span m="2984220">what's</span> <span m="2984440">the</span> <span m="2984530">expected</span> <span m="2984950">number</span> <span m="2985170">of</span> <span m="2985260">winners?</span> <span m="2988430">What</span> <span m="2988570">is</span> <span m="2988700">it?</span> <span m="2991830">We've</span> <span m="2992290">got 10</span> <span m="2992620">million</span> <span m="2997830">over</span> <span m="2998250">10,000,</span> <span m="3000680">right?</span> <span m="3000980">Because</span> <span m="3001190">what</span> <span m="3001300">I'm</span> <span m="3001430">doing</span> <span m="3001750">here</span> <span m="3002040">is</span> <span m="3002140">the</span> <span m="3002210">number</span> <span m="3002560">of</span> <span m="3002650">winners,</span> <span m="3003100">T,</span> <span m="3004380">is</span> <span m="3004580">going</span> <span m="3004720">to</span> <span m="3004780">be</span> <span m="3004850">the</span> <span m="3004970">sum</span> <span m="3005580">of</span> <span m="3005870">10</span> <span m="3006180">million</span> <span m="3006510">indicator</span> <span m="3007090">variables.</span> <span m="3014060">And</span> <span m="3014220">the</span> <span m="3014280">probability</span> <span m="3014980">that</span> <span m="3015160">any</span> <span m="3015360">one</span> <span m="3015500">of</span> <span m="3015570">these</span> <span m="3015720">guys</span> <span m="3016010">wins</span> <span m="3017450">is</span> <span m="3017670">1</span> <span m="3017980">in</span> <span m="3018180">10,000.</span> <span m="3021840">So</span> <span m="3022120">the</span> <span m="3022260">expected</span> <span m="3022820">number</span> <span m="3023090">of</span> <span m="3023190">winners</span> <span m="3023670">is</span> <span m="3023980">1</span> <span m="3024250">in</span> <span m="3024350">10,000</span> <span m="3025000">added</span> <span m="3025840">10</span> <span m="3026060">million</span> <span m="3026350">times,</span> <span m="3027990">which</span> <span m="3028190">is</span> <span m="3028320">this.</span> </p>
<p><span m="3032470">Is</span> <span m="3032550">that</span> <span m="3032700">OK?</span> <span m="3032990">Everybody</span> <span m="3033170">should</span> <span m="3033310">be</span> <span m="3033410">really</span> <span m="3033860">familiar</span> <span m="3034320">with</span> <span m="3034450">how</span> <span m="3034610">to</span> <span m="3035570">whip</span> <span m="3035850">these</span> <span m="3036080">things</span> <span m="3036360">out.</span> <span m="3036560">This</span> <span m="3036730">for</span> <span m="3036900">sure</span> <span m="3037320">will</span> <span m="3037480">have</span> <span m="3037880">probably</span> <span m="3038160">at</span> <span m="3038280">least</span> <span m="3038480">a</span> <span m="3038520">couple</span> <span m="3038790">questions</span> <span m="3039250">where</span> <span m="3039330">you're going to</span> <span m="3039420">need</span> <span m="3039650">to</span> <span m="3039720">be</span> <span m="3039790">able</span> <span m="3039910">to</span> <span m="3039970">do</span> <span m="3040150">that</span> <span m="3040340">kind</span> <span m="3040510">of</span> <span m="3040570">stuff</span> <span m="3041670">on</span> <span m="3041850">the</span> <span m="3041930">final.</span> </p>
<p><span m="3046710">All</span> <span m="3046970">right,</span> <span m="3047210">say</span> <span m="3047430">I</span> <span m="3047480">want</span> <span m="3047700">to</span> <span m="3047760">know</span> <span m="3047860">the</span> <span m="3047940">probability</span> <span m="3048720">of</span> <span m="3048890">getting</span> <span m="3050390">at</span> <span m="3050530">least</span> <span m="3050800">2,000</span> <span m="3051430">winners,</span> <span m="3064970">and</span> <span m="3065130">I</span> <span m="3065190">want</span> <span m="3065360">to</span> <span m="3065420">upper</span> <span m="3065640">bound</span> <span m="3065940">that</span> <span m="3067460">just</span> <span m="3067850">with</span> <span m="3067950">the</span> <span m="3068010">information</span> <span m="3068540">I've</span> <span m="3068640">given</span> <span m="3068890">you.</span> <span m="3072320">Well,</span> <span m="3074050">any</span> <span m="3074270">thoughts</span> <span m="3074840">about</span> <span m="3075850">an</span> <span m="3075980">upper</span> <span m="3076140">bound?</span> </p>
<p><span m="3077920">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3079260">PROFESSOR: What's</span> <span m="3079440">that?</span> </p>
<p><span m="3080364">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3082360">PROFESSOR: Yeah,</span> <span m="3082740">that's</span> <span m="3083000">a</span> <span m="3083070">good</span> <span m="3083660">upper</span> <span m="3083850">bound.</span> <span m="3085970">What</span> <span m="3086150">did</span> <span m="3086300">you</span> <span m="3086370">have</span> <span m="3086500">to</span> <span m="3086820">assume to</span> <span m="3087140">get</span> <span m="3087410">there?</span> <span m="3091330">e</span> <span m="3091630">to the</span> <span m="3091740">minus</span> <span m="3092090">380</span> <span m="3092470">is</span> <span m="3092580">a</span> <span m="3092640">great</span> <span m="3092940">bound.</span> <span m="3093680">Because</span> <span m="3094000">you're</span> <span m="3094100">going</span> <span m="3094220">to</span> <span m="3094280">plug</span> <span m="3094725">in</span> <span m="3095800">expected</span> <span m="3096520">value</span> <span m="3097220">is</span> <span m="3097920">1,000.</span> <span m="3099070">And</span> <span m="3099760">we're</span> <span m="3099870">asking</span> <span m="3100120">for</span> <span m="3100230">more</span> <span m="3100440">than</span> <span m="3100570">twice</span> <span m="3100940">the</span> <span m="3101020">expected</span> <span m="3101430">value.</span> <span m="3101810">So</span> <span m="3102000">it's</span> <span m="3102150">e to</span> <span m="3102420">the</span> <span m="3102450">minus</span> <span m="3102760">0.38</span> <span m="3103270">times</span> <span m="3103570">1,000.</span> <span m="3104640">And</span> <span m="3104800">that</span> <span m="3104990">for</span> <span m="3105100">sure</span> <span m="3105660">is--</span> <span m="3109670">so</span> <span m="3109860">you</span> <span m="3109970">computed</span> <span m="3110440">this.</span> <span m="3110790">And</span> <span m="3110910">that</span> <span m="3111160">equals</span> <span m="3111690">e</span> <span m="3111850">to</span> <span m="3112010">the</span> <span m="3112080">minus</span> <span m="3112480">380.</span> <span m="3114890">So</span> <span m="3115070">that's</span> <span m="3115280">really</span> <span m="3115530">small.</span> </p>
<p><span m="3116280">But</span> <span m="3116490">what</span> <span m="3116610">did</span> <span m="3116730">you</span> <span m="3116820">have</span> <span m="3116940">to</span> <span m="3117410">assume</span> <span m="3117880">to</span> <span m="3118350">apply</span> <span m="3119170">Chernoff?</span> <span m="3121670">Mutual</span> <span m="3122080">independence.</span> <span m="3122710">Mutual</span> <span m="3122780">independence</span> <span m="3123550">of</span> <span m="3123710">what?</span> </p>
<p><span m="3125020">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3126880">PROFESSOR: The</span> <span m="3127010">numbers</span> <span m="3127490">people</span> <span m="3127800">picked.</span> <span m="3129760">And</span> <span m="3129960">we</span> <span m="3130080">already</span> <span m="3130340">know,</span> <span m="3130740">if</span> <span m="3130860">people</span> <span m="3131200">are</span> <span m="3131280">picking</span> <span m="3131620">numbers,</span> <span m="3133390">they</span> <span m="3133480">don't</span> <span m="3133840">tend to</span> <span m="3133940">be</span> <span m="3134150">mutually</span> <span m="3134260">independent.</span> <span m="3135040">They</span> <span m="3135150">tend</span> <span m="3135450">to</span> <span m="3135520">gang</span> <span m="3135890">up.</span> <span m="3138030">But</span> <span m="3138200">if</span> <span m="3138350">you</span> <span m="3138510">had</span> <span m="3138680">a</span> <span m="3138710">computer</span> <span m="3139200">picking</span> <span m="3139510">the</span> <span m="3139580">numbers</span> <span m="3140010">randomly</span> <span m="3140500">and</span> <span m="3140590">mutually</span> <span m="3140940">independently,</span> <span m="3142260">then</span> <span m="3143380">you</span> <span m="3143530">would</span> <span m="3143640">be</span> <span m="3143860">e to</span> <span m="3144120">the</span> <span m="3144180">minus</span> <span m="3144510">380</span> <span m="3147010">by</span> <span m="3147240">Chernoff</span> <span m="3148650">if</span> <span m="3150380">mutually</span> <span m="3150940">independent</span> <span m="3153260">picks.</span> </p>
<p><span m="3157180">Everybody</span> <span m="3157530">see</span> <span m="3157800">why</span> <span m="3158000">we</span> <span m="3158110">did</span> <span m="3158290">that?</span> <span m="3159270">Because</span> <span m="3159510">it's</span> <span m="3159650">a</span> <span m="3159720">probability</span> <span m="3160150">of</span> <span m="3160230">twice</span> <span m="3160600">your</span> <span m="3160700">expectation.</span> <span m="3163320">The</span> <span m="3163430">total</span> <span m="3163770">number</span> <span m="3163980">of</span> <span m="3164060">winners</span> <span m="3164530">is</span> <span m="3164750">the</span> <span m="3164840">sum</span> <span m="3166040">of</span> <span m="3166270">10</span> <span m="3166680">million</span> <span m="3167440">indicator</span> <span m="3168080">variables.</span> <span m="3169640">And</span> <span m="3169790">indicator</span> <span m="3170220">variables</span> <span m="3170690">are</span> <span m="3170760">0</span> <span m="3171110">or</span> <span m="3171270">1.</span> <span m="3171660">So</span> <span m="3171810">they</span> <span m="3171910">fit</span> <span m="3172210">that</span> <span m="3172470">definition</span> <span m="3172990">up</span> <span m="3173130">there.</span> <span m="3176380">And</span> <span m="3176610">so</span> <span m="3177460">we</span> <span m="3177580">already</span> <span m="3177740">figured</span> <span m="3178040">out</span> <span m="3178760">z</span> <span m="3179540">is</span> <span m="3180280">at</span> <span m="3180750">least</span> <span m="3181020">0.38.</span> <span m="3183400">And</span> <span m="3183650">you're</span> <span m="3183740">multiplying</span> <span m="3184130">by</span> <span m="3184190">the</span> <span m="3184250">expected</span> <span m="3184580">value</span> <span m="3184780">of</span> <span m="3184840">1,000.</span> <span m="3185330">That's</span> <span m="3185460">e</span> <span m="3185570">to</span> <span m="3185650">the</span> <span m="3185710">minus</span> <span m="3185990">380,</span> <span m="3188710">so</span> <span m="3188870">very,</span> <span m="3189250">very</span> <span m="3189340">unlikely.</span> </p>
<p><span m="3190680">What</span> <span m="3190930">if</span> <span m="3191020">they</span> <span m="3191140">weren't</span> <span m="3191450">mutually</span> <span m="3191830">independent?</span> <span m="3192360">Can</span> <span m="3192470">you</span> <span m="3192550">say</span> <span m="3192840">anything</span> <span m="3193750">about</span> <span m="3194100">this,</span> <span m="3196530">anything</span> <span m="3196940">at</span> <span m="3197020">all</span> <span m="3197220">better</span> <span m="3197480">than</span> <span m="3198150">1,</span> <span m="3199630">which</span> <span m="3199830">we</span> <span m="3199900">know</span> <span m="3200020">for</span> <span m="3200180">any</span> <span m="3200310">probability?</span> <span m="3201202">Yeah?</span> </p>
<p><span m="3203090">AUDIENCE: It's</span> <span m="3203440">possible</span> <span m="3203870">that</span> <span m="3204910">everyone</span> <span m="3205310">chose</span> <span m="3205530">the</span> <span m="3205590">same</span> <span m="3205770">numbers.</span> </p>
<p><span m="3209020">PROFESSOR: Yes,</span> <span m="3209660">everyone</span> <span m="3210030">could have</span> <span m="3210200">chosen</span> <span m="3210490">the</span> <span m="3210590">same</span> <span m="3210890">number.</span> <span m="3211290">But</span> <span m="3212850">that</span> <span m="3213080">number</span> <span m="3213460">only</span> <span m="3213660">comes</span> <span m="3214010">up</span> <span m="3215640">with</span> <span m="3215770">a</span> <span m="3215810">1</span> <span m="3216010">in</span> <span m="3216080">10,000</span> <span m="3216660">chance.</span> <span m="3220950">So</span> <span m="3222010">you</span> <span m="3222200">can</span> <span m="3222500">say</span> <span m="3222710">something.</span> </p>
<p><span m="3223620">AUDIENCE: You can use</span> <span m="3224055">Markov.</span> </p>
<p><span m="3224490">PROFESSOR: Use</span> <span m="3224750">Markov.</span> <span m="3225350">What</span> <span m="3225510">does</span> <span m="3225610">Markov</span> <span m="3226020">give</span> <span m="3226200">you?</span> <span m="3234700">What</span> <span m="3234840">does</span> <span m="3234950">Markov</span> <span m="3235370">give</span> <span m="3235550">you?</span> <span m="3238480">1/2,</span> <span m="3239580">yeah.</span> <span m="3240480">Because</span> <span m="3241210">you've</span> <span m="3241370">got</span> <span m="3241690">the</span> <span m="3241790">expected</span> <span m="3242450">value</span> <span m="3244830">is</span> <span m="3245050">1,000</span> <span m="3247420">divided</span> <span m="3247880">by</span> <span m="3248640">the</span> <span m="3248740">bound</span> <span m="3249120">threshold,</span> <span m="3249590">is</span> <span m="3249690">2,000,</span> <span m="3251400">is</span> <span m="3251570">1/2</span> <span m="3252480">by</span> <span m="3252660">Markov.</span> <span m="3253220">And</span> <span m="3253510">that</span> <span m="3253840">holds</span> <span m="3254090">true</span> <span m="3254280">without</span> <span m="3254650">any</span> <span m="3254880">independence</span> <span m="3255420">assumption.</span> </p>
<p><span m="3259230">Now,</span> <span m="3259930">there</span> <span m="3260120">is</span> <span m="3260200">an</span> <span m="3260280">enormous</span> <span m="3260730">difference</span> <span m="3261130">between</span> <span m="3261420">1/2</span> <span m="3262450">and</span> <span m="3262590">e</span> <span m="3262720">to</span> <span m="3262780">the</span> <span m="3262840">minus</span> <span m="3263160">380.</span> <span m="3264810">Independence</span> <span m="3265640">really</span> <span m="3266110">makes</span> <span m="3266320">a</span> <span m="3266380">huge</span> <span m="3267090">difference</span> <span m="3268140">in</span> <span m="3268270">the</span> <span m="3268340">bound</span> <span m="3268640">you</span> <span m="3268720">can</span> <span m="3268870">compute.</span> <span m="3273700">OK,</span> <span m="3276020">now</span> <span m="3276190">there's</span> <span m="3276360">another</span> <span m="3276650">way</span> <span m="3276790">we</span> <span m="3276900">could've</span> <span m="3277130">gone</span> <span m="3277300">about</span> <span m="3277610">this.</span> <span m="3279050">What</span> <span m="3279300">kind</span> <span m="3279520">of</span> <span m="3279590">distribution</span> <span m="3280220">does</span> <span m="3280370">T</span> <span m="3280580">have</span> <span m="3282550">in</span> <span m="3282680">this</span> <span m="3282840">case?</span> <span m="3285952">It's</span> <span m="3286400">binomial.</span> <span m="3288160">Because</span> <span m="3288690">it's</span> <span m="3288860">the</span> <span m="3288950">sum</span> <span m="3289480">of</span> <span m="3289600">indicator</span> <span m="3290320">random</span> <span m="3290630">variables,</span> <span m="3291060">0,</span> <span m="3291370">1's.</span> <span m="3292370">Each</span> <span m="3292590">of</span> <span m="3292660">these</span> <span m="3292830">is</span> <span m="3292940">0,</span> <span m="3293240">1.</span> <span m="3294160">And</span> <span m="3294320">they're</span> <span m="3294430">all</span> <span m="3294620">the</span> <span m="3294680">same</span> <span m="3295000">distribution.</span> <span m="3295920">There's</span> <span m="3296090">a</span> <span m="3296140">1</span> <span m="3296350">in</span> <span m="3296440">10,000</span> <span m="3296980">chance</span> <span m="3297260">of</span> <span m="3297340">winning</span> <span m="3297950">for</span> <span m="3298120">each</span> <span m="3298620">one</span> <span m="3298810">of</span> <span m="3298880">them.</span> <span m="3300160">So</span> <span m="3300240">it's</span> <span m="3300350">a</span> <span m="3300390">binomial.</span> </p>
<p><span m="3301520">So</span> <span m="3301710">we</span> <span m="3301820">could</span> <span m="3302040">have</span> <span m="3302140">gone</span> <span m="3302350">back</span> <span m="3302600">and</span> <span m="3302730">used</span> <span m="3302970">the</span> <span m="3303050">formulas</span> <span m="3303700">we</span> <span m="3303810">had</span> <span m="3304000">for</span> <span m="3304060">the</span> <span m="3304130">binomial</span> <span m="3304570">distribution,</span> <span m="3305860">plugged</span> <span m="3306190">it</span> <span m="3306270">all</span> <span m="3306520">in,</span> <span m="3308260">and</span> <span m="3309380">we'd</span> <span m="3309590">have</span> <span m="3309680">gotten</span> <span m="3309950">something</span> <span m="3310540">pretty</span> <span m="3310760">similar</span> <span m="3311140">here.</span> <span m="3312030">But</span> <span m="3312200">Chernoff</span> <span m="3312450">is</span> <span m="3312690">so</span> <span m="3312860">much</span> <span m="3313080">easier.</span> <span m="3313580">Remember</span> <span m="3313840">that</span> <span m="3314030">pain</span> <span m="3314290">we</span> <span m="3314370">would</span> <span m="3314540">go</span> <span m="3314720">through</span> <span m="3315380">with</span> <span m="3315670">a</span> <span m="3315720">binomial</span> <span m="3316430">distribution,</span> <span m="3317040">the</span> <span m="3317110">approximation,</span> <span m="3318030">Stirling's</span> <span m="3318580">formula,</span> <span m="3319130">[INAUDIBLE]</span> <span m="3319830">whatever,</span> <span m="3321090">the</span> <span m="3321180">factorials</span> <span m="3321495">and</span> <span m="3321940">stuff?</span> <span m="3322285">And</span> <span m="3322630">that's</span> <span m="3322860">a</span> <span m="3322910">nightmare.</span> </p>
<p><span m="3324410">This</span> <span m="3324630">was</span> <span m="3324790">easy.</span> <span m="3325510">e to</span> <span m="3325730">the</span> <span m="3325790">minus</span> <span m="3326100">380</span> <span m="3326570">was</span> <span m="3326750">very</span> <span m="3327030">easy</span> <span m="3327260">to</span> <span m="3327330">compute.</span> <span m="3328490">And</span> <span m="3328610">really</span> <span m="3329770">at</span> <span m="3329890">that</span> <span m="3330070">point it</span> <span m="3330300">doesn't</span> <span m="3330520">matter</span> <span m="3330730">if</span> <span m="3330800">it's</span> <span m="3331130">minus</span> <span m="3331470">381</span> <span m="3331710">or</span> <span m="3332040">minus</span> <span m="3332310">382</span> <span m="3332860">or</span> <span m="3332930">whatever.</span> <span m="3333950">Because</span> <span m="3334110">it's</span> <span m="3334260">really</span> <span m="3334470">small.</span> <span m="3335870">So</span> <span m="3336060">often,</span> <span m="3336470">even</span> <span m="3336660">when</span> <span m="3336770">you</span> <span m="3336830">have</span> <span m="3336930">a</span> <span m="3336970">binomial</span> <span m="3337500">distribution,</span> <span m="3339020">well,</span> <span m="3339250">Chernoff</span> <span m="3339680">will</span> <span m="3339810">apply.</span> <span m="3340250">And</span> <span m="3340340">that's</span> <span m="3340510">a</span> <span m="3340570">great</span> <span m="3340800">way</span> <span m="3340900">to</span> <span m="3340970">go.</span> <span m="3342560">Because</span> <span m="3342760">it</span> <span m="3342840">gives</span> <span m="3343010">you</span> <span m="3343080">good</span> <span m="3343250">bounds</span> <span m="3344610">generally.</span> </p>
<p><span m="3346180">All</span> <span m="3346340">right,</span> <span m="3346960">let's</span> <span m="3347460">figure</span> <span m="3347800">out</span> <span m="3348230">the</span> <span m="3348320">probability</span> <span m="3349170">of</span> <span m="3350730">at</span> <span m="3350890">least</span> <span m="3351160">1,100</span> <span m="3352230">winners</span> <span m="3353400">instead</span> <span m="3353660">of</span> <span m="3353730">1,000.</span> <span m="3356600">So</span> <span m="3356830">let's</span> <span m="3357070">look</span> <span m="3357270">at</span> <span m="3357680">the</span> <span m="3357750">probability</span> <span m="3359150">of</span> <span m="3359620">at</span> <span m="3359740">least</span> <span m="3360860">100</span> <span m="3361390">extra</span> <span m="3361710">winners</span> <span m="3362110">over</span> <span m="3362350">what</span> <span m="3362520">we</span> <span m="3362630">expect</span> <span m="3363140">out</span> <span m="3363300">of</span> <span m="3363410">10</span> <span m="3363700">million.</span> <span m="3365520">We've</span> <span m="3365660">got</span> <span m="3365840">10</span> <span m="3366120">million</span> <span m="3366420">people.</span> <span m="3366790">You</span> <span m="3366900">expect</span> <span m="3367260">1,000.</span> <span m="3367730">We're</span> <span m="3367820">going</span> <span m="3367940">to</span> <span m="3368020">analyze</span> <span m="3368500">the</span> <span m="3368550">probability</span> <span m="3368990">of</span> <span m="3369050">1,100.</span> <span m="3370830">What's</span> <span m="3371060">c</span> <span m="3371430">in</span> <span m="3371530">this</span> <span m="3371710">case?</span> <span m="3372600">We're</span> <span m="3372740">going to</span> <span m="3372870">use</span> <span m="3372990">Chernoff.</span> </p>
<p><span m="3375530">1.1.</span> <span m="3377550">So</span> <span m="3377680">this</span> <span m="3378070">is</span> <span m="3378190">1.1</span> <span m="3379020">times</span> <span m="3379340">1,000.</span> <span m="3381090">And</span> <span m="3381240">that</span> <span m="3381430">means</span> <span m="3381780">that</span> <span m="3381930">z</span> <span m="3383990">is</span> <span m="3385190">1.1</span> <span m="3386090">times</span> <span m="3386410">the</span> <span m="3386470">natural</span> <span m="3386850">log</span> <span m="3387150">of</span> <span m="3387250">1.1</span> <span m="3388560">plus</span> <span m="3388860">1</span> <span m="3389180">minus</span> <span m="3389660">1.1.</span> <span m="3391160">And</span> <span m="3391320">that</span> <span m="3391490">is</span> <span m="3391620">close</span> <span m="3392050">to</span> <span m="3392430">but</span> <span m="3392560">at</span> <span m="3392670">least</span> <span m="3393580">0.0048.</span> <span m="3399020">So</span> <span m="3399480">this</span> <span m="3399660">probability</span> <span m="3401200">is</span> <span m="3401450">at</span> <span m="3401580">most,</span> <span m="3402090">by</span> <span m="3402200">Chernoff,</span> <span m="3402840">e</span> <span m="3403090">to</span> <span m="3403170">the</span> <span m="3403240">minus</span> <span m="3404020">0.0048</span> <span m="3406830">times</span> <span m="3408680">the</span> <span m="3408790">expected</span> <span m="3409460">number</span> <span m="3409700">of</span> <span m="3409790">winners</span> <span m="3410150">is</span> <span m="3410280">1,000.</span> <span m="3413010">So</span> <span m="3413310">that</span> <span m="3413580">is</span> <span m="3415270">e</span> <span m="3415620">to</span> <span m="3415690">the</span> <span m="3415760">minus</span> <span m="3416330">4.8,</span> <span m="3418900">which</span> <span m="3419040">is</span> <span m="3419210">less</span> <span m="3419510">than</span> <span m="3419730">1%,</span> <span m="3419930">1 in</span> <span m="3420330">100.</span> </p>
<p><span m="3424020">So</span> <span m="3424130">that's</span> <span m="3424270">pretty</span> <span m="3424450">powerful.</span> <span m="3424900">It</span> <span m="3424970">says,</span> <span m="3425600">you've</span> <span m="3425730">got</span> <span m="3425920">10</span> <span m="3426300">million</span> <span m="3426610">people</span> <span m="3426930">who</span> <span m="3427030">could</span> <span m="3427190">win.</span> <span m="3428720">The</span> <span m="3428840">chance</span> <span m="3429330">of</span> <span m="3429420">even</span> <span m="3429640">having</span> <span m="3429860">100</span> <span m="3430410">more</span> <span m="3431050">than</span> <span m="3431230">the</span> <span m="3431340">1,000</span> <span m="3431740">you</span> <span m="3431840">expect</span> <span m="3433250">is</span> <span m="3433680">1%</span> <span m="3434290">chance</span> <span m="3434610">at</span> <span m="3434700">most--</span> <span m="3436680">very,</span> <span m="3437180">very</span> <span m="3437380">powerful.</span> <span m="3437880">It</span> <span m="3437960">says</span> <span m="3438210">you</span> <span m="3438470">really</span> <span m="3438930">expect</span> <span m="3439340">to</span> <span m="3439410">get</span> <span m="3439820">really</span> <span m="3440220">close</span> <span m="3441060">to</span> <span m="3441200">the</span> <span m="3441280">mean</span> <span m="3442530">in</span> <span m="3442670">this</span> <span m="3442790">situation.</span> </p>
<p><span m="3445660">OK,</span> <span m="3445880">a</span> <span m="3445930">lot</span> <span m="3446200">better--</span> <span m="3447260">Markov</span> <span m="3447820">here</span> <span m="3448430">gives</span> <span m="3448670">you,</span> <span m="3449280">what,</span> <span m="3450390">1,000</span> <span m="3450990">over</span> <span m="3451190">1,100.</span> <span m="3452380">It</span> <span m="3452700">says</span> <span m="3452920">your</span> <span m="3453020">probability</span> <span m="3453540">could</span> <span m="3453670">be</span> <span m="3453770">90%</span> <span m="3454430">or</span> <span m="3454500">something--</span> <span m="3455430">not</span> <span m="3455690">very</span> <span m="3455930">useful.</span> <span m="3456290">Chebyshev</span> <span m="3456840">won't</span> <span m="3457030">give</span> <span m="3457180">you</span> <span m="3457260">much</span> <span m="3457590">here</span> <span m="3457780">either.</span> <span m="3458760">So</span> <span m="3458830">if</span> <span m="3459760">you're</span> <span m="3459910">in a</span> <span m="3459960">situation to</span> <span m="3460410">apply</span> <span m="3460670">Chernoff,</span> <span m="3461730">always</span> <span m="3462140">go</span> <span m="3462310">there.</span> <span m="3462915">It</span> <span m="3463280">gives</span> <span m="3463470">you</span> <span m="3463530">the</span> <span m="3463610">best</span> <span m="3463860">bounds.</span> <span m="3464800">Any</span> <span m="3464990">questions?</span> <span m="3468100">This</span> <span m="3468310">of</span> <span m="3468370">course</span> <span m="3468560">is</span> <span m="3468620">why</span> <span m="3468780">computer</span> <span m="3469150">scientists</span> <span m="3469840">use</span> <span m="3470040">it</span> <span m="3470130">all</span> <span m="3470240">the</span> <span m="3470310">time.</span> </p>
<p><span m="3474010">OK,</span> <span m="3475570">actually,</span> <span m="3475870">before</span> <span m="3476060">I do</span> <span m="3476160">more</span> <span m="3476340">examples,</span> <span m="3476990">let</span> <span m="3477140">me</span> <span m="3477880">prove</span> <span m="3478560">the</span> <span m="3478640">theorem</span> <span m="3479020">in</span> <span m="3479090">a</span> <span m="3479130">special</span> <span m="3479480">case</span> <span m="3479770">to</span> <span m="3479860">give</span> <span m="3479990">you</span> <span m="3480080">a</span> <span m="3480120">feel</span> <span m="3480490">for</span> <span m="3481690">what's</span> <span m="3481920">involved.</span> <span m="3483050">The</span> <span m="3483170">full</span> <span m="3483430">proof</span> <span m="3483750">is</span> <span m="3483870">in</span> <span m="3483950">the</span> <span m="3484030">text.</span> <span m="3485950">I'm</span> <span m="3486120">going</span> <span m="3486240">to</span> <span m="3486320">prove</span> <span m="3486730">it</span> <span m="3486800">in</span> <span m="3486890">the</span> <span m="3486960">special</span> <span m="3487430">case</span> <span m="3487960">where</span> <span m="3488720">the</span> <span m="3488860">Tj</span> <span m="3489680">are</span> <span m="3489940">0,</span> <span m="3490340">1.</span> <span m="3491090">So</span> <span m="3491270">they're</span> <span m="3491410">indicator</span> <span m="3491930">random</span> <span m="3492170">variables.</span> </p>
<p><span m="3492860">But</span> <span m="3493110">they</span> <span m="3493230">don't</span> <span m="3493590">have</span> <span m="3493810">to</span> <span m="3493920">have</span> <span m="3494190">the</span> <span m="3494270">same</span> <span m="3494570">distribution.</span> <span m="3496020">So</span> <span m="3496140">it's</span> <span m="3496260">still</span> <span m="3496590">more</span> <span m="3496830">general</span> <span m="3497260">than</span> <span m="3497390">you</span> <span m="3497480">get</span> <span m="3497730">with</span> <span m="3497840">a</span> <span m="3497880">binomial</span> <span m="3498370">distribution.</span> <span m="3500930">All</span> <span m="3501070">right,</span> <span m="3501170">so</span> <span m="3501280">we're</span> <span m="3501370">going</span> <span m="3501490">to</span> <span m="3501550">do</span> <span m="3501660">a</span> <span m="3501730">proof</span> <span m="3504740">of</span> <span m="3504960">Chernoff</span> <span m="3508420">for</span> <span m="3508560">the</span> <span m="3508650">special</span> <span m="3509040">case</span> <span m="3511800">where</span> <span m="3512060">the</span> <span m="3512160">Tj</span> <span m="3514440">are</span> <span m="3515040">either</span> <span m="3515430">0</span> <span m="3515810">or</span> <span m="3515890">1.</span> <span m="3517560">So</span> <span m="3517730">they're</span> <span m="3517870">indicator</span> <span m="3518360">variables.</span> </p>
<p><span m="3522980">OK,</span> <span m="3523390">so</span> <span m="3523540">the</span> <span m="3523610">first</span> <span m="3523900">step</span> <span m="3524190">is</span> <span m="3524300">going</span> <span m="3524420">to</span> <span m="3524500">seem</span> <span m="3525270">pretty</span> <span m="3525480">mysterious.</span> <span m="3526140">But</span> <span m="3526250">we've</span> <span m="3526380">been</span> <span m="3526540">doing</span> <span m="3526830">something</span> <span m="3527200">like</span> <span m="3527420">it</span> <span m="3527530">all</span> <span m="3527700">day.</span> <span m="3532090">I'm</span> <span m="3532270">trying</span> <span m="3532550">to</span> <span m="3532620">compute</span> <span m="3532970">the</span> <span m="3533040">probability</span> <span m="3533530">T</span> <span m="3533930">is</span> <span m="3534080">bigger</span> <span m="3534320">than</span> <span m="3534620">c</span> <span m="3534910">times</span> <span m="3535200">its</span> <span m="3535330">expectation.</span> <span m="3537670">Well,</span> <span m="3537930">what</span> <span m="3538020">I'm</span> <span m="3538080">going</span> <span m="3538200">to</span> <span m="3538260">do</span> <span m="3538800">is</span> <span m="3539080">exponentiate</span> <span m="3540490">both</span> <span m="3540760">of</span> <span m="3540840">these</span> <span m="3541060">guys</span> <span m="3542500">and</span> <span m="3542870">compute</span> <span m="3543480">the</span> <span m="3543550">probability</span> <span m="3543960">that</span> <span m="3544120">c</span> <span m="3544460">to</span> <span m="3544550">the</span> <span m="3544660">T</span> <span m="3546230">is</span> <span m="3546460">at</span> <span m="3546570">least</span> <span m="3546970">c</span> <span m="3547440">to</span> <span m="3547590">the</span> <span m="3547700">c</span> <span m="3548193">times</span> <span m="3548686">the</span> <span m="3549180">expected</span> <span m="3549640">value</span> <span m="3550102">of T.</span> </p>
<p><span m="3553120">Now,</span> <span m="3554520">this</span> <span m="3554720">is</span> <span m="3554820">not</span> <span m="3555230">the</span> <span m="3555310">first</span> <span m="3555650">thing</span> <span m="3555780">you'd</span> <span m="3555890">expect</span> <span m="3556290">to</span> <span m="3556360">do,</span> <span m="3556650">probably,</span> <span m="3557100">if</span> <span m="3557190">you were</span> <span m="3557330">trying</span> <span m="3557540">to</span> <span m="3557630">prove</span> <span m="3557890">this.</span> <span m="3558580">So</span> <span m="3558740">it's</span> <span m="3558850">one</span> <span m="3558990">of</span> <span m="3559050">those</span> <span m="3559220">divine</span> <span m="3559650">insights</span> <span m="3560600">that</span> <span m="3561120">you'd</span> <span m="3561280">make</span> <span m="3561530">this</span> <span m="3561680">step.</span> <span m="3564920">And</span> <span m="3565090">then</span> <span m="3565340">I'm</span> <span m="3565440">going</span> <span m="3565560">to</span> <span m="3565620">apply</span> <span m="3565870">Markov,</span> <span m="3566760">like</span> <span m="3566970">we've</span> <span m="3567070">been</span> <span m="3567170">doing</span> <span m="3567400">all</span> <span m="3567660">day,</span> <span m="3568200">to</span> <span m="3568320">this.</span> <span m="3570960">Now,</span> <span m="3571250">since</span> <span m="3571530">T</span> <span m="3571710">is</span> <span m="3571860">positive</span> <span m="3572350">and</span> <span m="3572460">c</span> <span m="3572700">is</span> <span m="3572850">positive,</span> <span m="3574390">these</span> <span m="3574640">are</span> <span m="3574750">equal.</span> <span m="3575820">And</span> <span m="3576720">this</span> <span m="3576950">is</span> <span m="3577060">never</span> <span m="3577330">non-negative.</span> </p>
<p><span m="3579520">So</span> <span m="3579710">now</span> <span m="3580010">by</span> <span m="3580150">Markov,</span> <span m="3581950">this</span> <span m="3582230">is</span> <span m="3582330">simply</span> <span m="3582750">upper</span> <span m="3582860">bounded</span> <span m="3583250">by</span> <span m="3583550">the</span> <span m="3583670">expected</span> <span m="3584400">value</span> <span m="3584740">of</span> <span m="3584810">that,</span> <span m="3587380">expected</span> <span m="3587800">value</span> <span m="3588030">of</span> <span m="3588120">c</span> <span m="3588310">to</span> <span m="3588390">the</span> <span m="3588490">T,</span> <span m="3590190">divided</span> <span m="3590650">by</span> <span m="3590870">this.</span> <span m="3597130">And</span> <span m="3597260">that's</span> <span m="3597420">by</span> <span m="3597550">Markov.</span> <span m="3598950">So</span> <span m="3599110">everything</span> <span m="3599490">we've</span> <span m="3599620">done</span> <span m="3599810">today</span> <span m="3600020">is</span> <span m="3600140">really</span> <span m="3600350">Markov</span> <span m="3600650">in</span> <span m="3600890">disguise.</span> </p>
<p><span m="3605500">Any</span> <span m="3605680">questions</span> <span m="3606520">so</span> <span m="3606710">far?</span> <span m="3607790">You</span> <span m="3607920">start</span> <span m="3608160">looking</span> <span m="3608450">at this,</span> <span m="3608570">you</span> <span m="3608660">go,</span> <span m="3608800">oh</span> <span m="3608860">my</span> <span m="3609000">god,</span> <span m="3609270">I</span> <span m="3609300">got</span> <span m="3609420">the</span> <span m="3609530">random</span> <span m="3609790">variable</span> <span m="3610170">and</span> <span m="3610230">the</span> <span m="3610340">exponent.</span> <span m="3611700">This</span> <span m="3611900">is</span> <span m="3612010">looking</span> <span m="3612230">like</span> <span m="3612420">a</span> <span m="3612460">nightmare.</span> <span m="3613150">What</span> <span m="3613340">is</span> <span m="3613450">the</span> <span m="3613540">expected</span> <span m="3613980">value</span> <span m="3614230">of</span> <span m="3614300">c</span> <span m="3614470">to</span> <span m="3614560">the</span> <span m="3614670">T,</span> <span m="3615040">and</span> <span m="3615270">this</span> <span m="3615400">kind</span> <span m="3615570">of</span> <span m="3615630">stuff?</span> <span m="3615880">But</span> <span m="3616000">we're</span> <span m="3616100">going</span> <span m="3616220">to</span> <span m="3616460">hack</span> <span m="3616760">through</span> <span m="3616980">it.</span> <span m="3618580">Because</span> <span m="3618760">it</span> <span m="3618870">gives</span> <span m="3619050">you</span> <span m="3619320">just</span> <span m="3619480">an</span> <span m="3619640">amazingly</span> <span m="3620160">powerful</span> <span m="3620570">result</span> <span m="3620900">when</span> <span m="3620990">you're</span> <span m="3621080">done.</span> </p>
<p><span m="3627880">All</span> <span m="3628100">right,</span> <span m="3628480">so</span> <span m="3628680">we've</span> <span m="3628800">got</span> <span m="3628980">to</span> <span m="3629050">evaluate</span> <span m="3630420">the</span> <span m="3630540">expected</span> <span m="3631110">value</span> <span m="3631510">of</span> <span m="3631590">c</span> <span m="3631920">to</span> <span m="3632010">the</span> <span m="3632120">T.</span> <span m="3634040">And</span> <span m="3634240">we're</span> <span m="3634310">going</span> <span m="3634440">to</span> <span m="3634510">use</span> <span m="3634880">the</span> <span m="3634950">fact</span> <span m="3635370">that</span> <span m="3635530">T</span> <span m="3636040">is</span> <span m="3636140">the</span> <span m="3636220">sum</span> <span m="3636620">of</span> <span m="3636720">the</span> <span m="3636790">Tj's.</span> <span m="3645510">And</span> <span m="3645670">that</span> <span m="3645870">means</span> <span m="3646290">that</span> <span m="3647060">c</span> <span m="3647550">to</span> <span m="3647620">the</span> <span m="3647690">T</span> <span m="3649600">equals</span> <span m="3650730">c</span> <span m="3651040">to</span> <span m="3651110">the</span> <span m="3651210">T1</span> <span m="3651870">times</span> <span m="3652110">c</span> <span m="3652440">to</span> <span m="3652510">the</span> <span m="3652580">T2</span> <span m="3654820">times</span> <span m="3654990">c</span> <span m="3655130">to</span> <span m="3655200">the</span> <span m="3655390">Tn.</span> </p>
<p><span m="3656770">The</span> <span m="3656970">weird</span> <span m="3657190">thing</span> <span m="3657320">about</span> <span m="3657520">this</span> <span m="3657680">proof</span> <span m="3657980">is</span> <span m="3658080">that</span> <span m="3658220">every</span> <span m="3658480">step</span> <span m="3659170">sort</span> <span m="3659410">of</span> <span m="3659480">makes</span> <span m="3659720">it</span> <span m="3659810">more</span> <span m="3660120">complicated</span> <span m="3660770">looking</span> <span m="3661710">until</span> <span m="3661990">we</span> <span m="3662080">get</span> <span m="3662210">to</span> <span m="3662300">the</span> <span m="3662440">end.</span> <span m="3663760">So</span> <span m="3663920">it's</span> <span m="3664030">one</span> <span m="3664200">of</span> <span m="3664270">those</span> <span m="3664520">that's</span> <span m="3664690">hard</span> <span m="3664950">to</span> <span m="3665030">figure</span> <span m="3665330">out</span> <span m="3665480">the</span> <span m="3665550">first</span> <span m="3665830">time.</span> <span m="3668750">All</span> <span m="3668930">right,</span> <span m="3669130">that</span> <span m="3669290">means</span> <span m="3669510">the</span> <span m="3669590">expected</span> <span m="3670140">value</span> <span m="3670450">of</span> <span m="3670520">c</span> <span m="3670730">to</span> <span m="3670830">the</span> <span m="3670930">T</span> <span m="3673470">is</span> <span m="3674030">the</span> <span m="3674160">expected</span> <span m="3674770">value</span> <span m="3676220">of</span> <span m="3676780">the</span> <span m="3676850">product</span> <span m="3677340">of</span> <span m="3677410">these</span> <span m="3677660">things.</span> <span m="3686310">Now</span> <span m="3688240">I'm</span> <span m="3688300">going</span> <span m="3688420">to</span> <span m="3688480">use</span> <span m="3688690">the</span> <span m="3688770">product</span> <span m="3689280">rule</span> <span m="3689860">for</span> <span m="3690040">expectation.</span> </p>
<p><span m="3699570">Now,</span> <span m="3699810">why</span> <span m="3700060">can</span> <span m="3700230">I</span> <span m="3700310">use</span> <span m="3700670">the</span> <span m="3700750">product</span> <span m="3701220">rule?</span> <span m="3701810">What</span> <span m="3702040">am</span> <span m="3702150">I</span> <span m="3702280">assuming</span> <span m="3702800">to</span> <span m="3702890">be</span> <span m="3702980">able</span> <span m="3703110">to</span> <span m="3703200">do</span> <span m="3703360">that?</span> <span m="3707160">That</span> <span m="3707640">they</span> <span m="3707910">are</span> <span m="3707950">mutually</span> <span m="3708420">independent,</span> <span m="3709150">that</span> <span m="3709300">the</span> <span m="3709400">c</span> <span m="3709820">to</span> <span m="3709890">the</span> <span m="3709970">Tj's</span> <span m="3710690">are</span> <span m="3710840">mutually</span> <span m="3711290">independent</span> <span m="3712040">of</span> <span m="3712180">each</span> <span m="3712390">other.</span> <span m="3713630">And</span> <span m="3713760">that</span> <span m="3713930">follows,</span> <span m="3714490">because</span> <span m="3714860">the</span> <span m="3714940">Tj's</span> <span m="3715740">are</span> <span m="3715810">mutually</span> <span m="3716230">independent.</span> </p>
<p><span m="3718580">So</span> <span m="3718790">if</span> <span m="3718880">a</span> <span m="3718920">bunch</span> <span m="3719160">of</span> <span m="3719220">random</span> <span m="3719440">variable are</span> <span m="3719830">mutually independent,</span> <span m="3720870">then</span> <span m="3721160">their</span> <span m="3721660">exponentiations</span> <span m="3722590">are</span> <span m="3722700">mutually</span> <span m="3723050">independent.</span> <span m="3725680">So</span> <span m="3725790">this</span> <span m="3726000">is</span> <span m="3726180">by</span> <span m="3727830">product</span> <span m="3728350">rule</span> <span m="3729740">for</span> <span m="3729900">expectation</span> <span m="3732340">and</span> <span m="3732620">mutual</span> <span m="3732940">independence.</span> <span m="3738800">OK,</span> <span m="3739190">so</span> <span m="3739330">now</span> <span m="3740130">we've</span> <span m="3740270">got</span> <span m="3740380">to</span> <span m="3740460">evaluate</span> <span m="3741150">the</span> <span m="3741220">expected</span> <span m="3741800">value</span> <span m="3743790">of</span> <span m="3744280">c</span> <span m="3744740">to</span> <span m="3744810">the</span> <span m="3744890">Tj.</span> </p>
<p><span m="3752580">And</span> <span m="3752800">this</span> <span m="3752950">is</span> <span m="3753060">where</span> <span m="3753230">we're</span> <span m="3753320">going</span> <span m="3753440">to</span> <span m="3753500">make</span> <span m="3753680">it</span> <span m="3753790">simpler</span> <span m="3754360">by</span> <span m="3754790">assuming</span> <span m="3755280">that</span> <span m="3755440">Tj</span> <span m="3755970">is</span> <span m="3756130">just</span> <span m="3756360">a</span> <span m="3756400">0,</span> <span m="3756780">1</span> <span m="3757650">random</span> <span m="3757990">variable.</span> <span m="3760070">So</span> <span m="3760470">the</span> <span m="3760540">simplification</span> <span m="3761370">comes</span> <span m="3761846">in here.</span> <span m="3767400">So</span> <span m="3767580">the</span> <span m="3767690">expected</span> <span m="3768270">value</span> <span m="3768850">of</span> <span m="3769330">Tj--</span> <span m="3771760">well,</span> <span m="3772120">there's</span> <span m="3772620">two</span> <span m="3772820">cases.</span> <span m="3773420">Tj</span> <span m="3773840">is</span> <span m="3773980">1,</span> <span m="3774400">or it's</span> <span m="3774600">0.</span> <span m="3775080">Because</span> <span m="3775240">we</span> <span m="3775300">made</span> <span m="3775550">this</span> <span m="3775670">simplification.</span> </p>
<p><span m="3777760">If</span> <span m="3777910">it's</span> <span m="3778050">1,</span> <span m="3779200">I</span> <span m="3779370">get</span> <span m="3779670">c</span> <span m="3779950">to</span> <span m="3780030">the</span> <span m="3780130">1--</span> <span m="3780800">ooh,</span> <span m="3781110">expected</span> <span m="3781260">value of</span> <span m="3781340">c</span> <span m="3781800">to</span> <span m="3781860">the</span> <span m="3781920">Tj.</span> <span m="3783010">Let's</span> <span m="3783380">get</span> <span m="3783600">that</span> <span m="3783790">right.</span> <span m="3787600">It</span> <span m="3787740">could</span> <span m="3787850">be</span> <span m="3787940">1,</span> <span m="3788310">in</span> <span m="3788390">which</span> <span m="3788560">case</span> <span m="3788770">I</span> <span m="3788840">get</span> <span m="3788980">a</span> <span m="3789010">contribution</span> <span m="3789650">of</span> <span m="3789730">c</span> <span m="3790060">to</span> <span m="3790120">the</span> <span m="3790190">1</span> <span m="3791650">times</span> <span m="3792160">the</span> <span m="3792230">probability</span> <span m="3793460">Tj</span> <span m="3793930">equals</span> <span m="3794300">1</span> <span m="3795560">plus</span> <span m="3796010">the</span> <span m="3796100">case</span> <span m="3796390">at</span> <span m="3796490">0.</span> <span m="3797590">So</span> <span m="3797760">I get</span> <span m="3797950">c</span> <span m="3798180">to</span> <span m="3798250">the</span> <span m="3798330">0</span> <span m="3799070">times</span> <span m="3799390">the</span> <span m="3799450">probability</span> <span m="3800010">Tj</span> <span m="3800510">is</span> <span m="3800610">0.</span> </p>
<p><span m="3804230">Well,</span> <span m="3804620">c</span> <span m="3804820">to the</span> <span m="3804980">1</span> <span m="3805230">is</span> <span m="3805360">just</span> <span m="3805650">c.</span> <span m="3812370">c</span> <span m="3812640">to</span> <span m="3812700">the</span> <span m="3812780">0</span> <span m="3813190">is</span> <span m="3813320">1.</span> <span m="3814620">And</span> <span m="3815260">I'm</span> <span m="3815390">going</span> <span m="3815530">to</span> <span m="3815620">rewrite</span> <span m="3816740">Tj</span> <span m="3817170">being</span> <span m="3817470">0</span> <span m="3817850">as</span> <span m="3817980">1</span> <span m="3818370">minus the</span> <span m="3818730">probability</span> <span m="3819160">Tj</span> <span m="3819490">is</span> <span m="3819620">1.</span> <span m="3825020">All</span> <span m="3825140">right,</span> <span m="3825420">this</span> <span m="3826470">equals</span> <span m="3826920">that.</span> <span m="3827530">And</span> <span m="3827620">of</span> <span m="3827680">course</span> <span m="3827920">the</span> <span m="3828000">1</span> <span m="3828400">cancels.</span> <span m="3829980">Now</span> <span m="3830180">I'm</span> <span m="3830240">going</span> <span m="3830360">to</span> <span m="3830680">collect</span> <span m="3831130">terms</span> <span m="3831490">here</span> <span m="3831750">to</span> <span m="3831820">get</span> <span m="3835270">1</span> <span m="3835470">plus</span> <span m="3836400">c</span> <span m="3836750">minus</span> <span m="3837150">1</span> <span m="3838440">times</span> <span m="3838790">the</span> <span m="3838870">probability</span> <span m="3839636">Tj</span> <span m="3840020">equals</span> <span m="3840502">1.</span> </p>
<p><span m="3846680">OK,</span> <span m="3848320">then</span> <span m="3848610">I'm</span> <span m="3848700">going</span> <span m="3848830">to</span> <span m="3849650">do</span> <span m="3849820">one</span> <span m="3850090">more</span> <span m="3850260">step</span> <span m="3850550">here.</span> <span m="3850720">This</span> <span m="3850900">is</span> <span m="3851020">1</span> <span m="3851360">plus</span> <span m="3852200">c</span> <span m="3852500">minus</span> <span m="3852880">1</span> <span m="3853930">times</span> <span m="3854210">the</span> <span m="3854300">expected</span> <span m="3854850">value</span> <span m="3855290">of</span> <span m="3855380">Tj.</span> <span m="3857730">Because</span> <span m="3858090">if</span> <span m="3858190">I</span> <span m="3858280">have</span> <span m="3858400">an</span> <span m="3858480">indicator</span> <span m="3859130">random</span> <span m="3859470">variable,</span> <span m="3861210">the</span> <span m="3861320">expected</span> <span m="3861840">value</span> <span m="3862290">is</span> <span m="3862420">the</span> <span m="3862500">same</span> <span m="3862890">as</span> <span m="3862980">the</span> <span m="3863060">probability</span> <span m="3863580">that</span> <span m="3863690">it's</span> <span m="3863830">1.</span> <span m="3864880">Because in</span> <span m="3865240">the</span> <span m="3865470">other</span> <span m="3865680">case</span> <span m="3865980">it's</span> <span m="3866060">0.</span> </p>
<p><span m="3868780">And</span> <span m="3868980">now</span> <span m="3869160">I'm</span> <span m="3869280">going</span> <span m="3869400">to</span> <span m="3869460">use</span> <span m="3869640">the</span> <span m="3869720">trick</span> <span m="3870010">from</span> <span m="3870180">last</span> <span m="3870560">time.</span> <span m="3871720">Remember</span> <span m="3872120">1</span> <span m="3872400">plus</span> <span m="3872740">x</span> <span m="3873110">is</span> <span m="3873280">always</span> <span m="3873660">at</span> <span m="3873750">most</span> <span m="3874070">e</span> <span m="3874210">to</span> <span m="3874270">the</span> <span m="3874400">x</span> <span m="3876420">from</span> <span m="3876500">last</span> <span m="3876810">time?</span> <span m="3877790">None</span> <span m="3877990">of</span> <span m="3878050">these</span> <span m="3878210">steps</span> <span m="3878500">is</span> <span m="3878620">obvious</span> <span m="3878970">why</span> <span m="3879110">we're</span> <span m="3879240">doing</span> <span m="3879490">them.</span> <span m="3879730">But</span> <span m="3880020">we're</span> <span m="3880210">going</span> <span m="3880340">to</span> <span m="3881110">do</span> <span m="3881310">them</span> <span m="3881400">anyway.</span> <span m="3883560">So</span> <span m="3884310">this</span> <span m="3884600">is</span> <span m="3884790">at</span> <span m="3884850">most</span> <span m="3887790">e</span> <span m="3888180">to</span> <span m="3888290">this,</span> <span m="3890470">c</span> <span m="3890760">minus</span> <span m="3891130">1</span> <span m="3892130">expected</span> <span m="3892560">value</span> <span m="3892920">of Tj.</span> <span m="3897110">Because</span> <span m="3897820">1</span> <span m="3898230">plus</span> <span m="3898460">anything</span> <span m="3898790">is</span> <span m="3898870">at</span> <span m="3898980">most</span> <span m="3899890">the</span> <span m="3900000">exponential</span> <span m="3900305">of</span> <span m="3900610">that.</span> </p>
<p><span m="3901730">And</span> <span m="3901870">I'm</span> <span m="3902030">doing</span> <span m="3902410">this</span> <span m="3902630">step</span> <span m="3903010">because</span> <span m="3904480">I</span> <span m="3904610">got</span> <span m="3904780">a</span> <span m="3904820">product</span> <span m="3905360">of</span> <span m="3905440">these</span> <span m="3905650">guys.</span> <span m="3906830">And</span> <span m="3906970">I</span> <span m="3907010">want</span> <span m="3907250">to</span> <span m="3907320">put</span> <span m="3907590">them</span> <span m="3907740">in</span> <span m="3907820">the</span> <span m="3907940">exponent</span> <span m="3908560">so</span> <span m="3908580">I</span> <span m="3908620">can</span> <span m="3908770">then</span> <span m="3908950">sum</span> <span m="3909240">them</span> <span m="3909430">so</span> <span m="3909570">it</span> <span m="3909650">gets</span> <span m="3909860">easy.</span> </p>
<p><span m="3923570">OK,</span> <span m="3926190">now</span> <span m="3926450">we</span> <span m="3926540">just</span> <span m="3926720">plug</span> <span m="3927810">this</span> <span m="3929550">back</span> <span m="3929910">in</span> <span m="3930070">here.</span> <span m="3932880">So</span> <span m="3933130">that</span> <span m="3933400">means</span> <span m="3933920">that</span> <span m="3934710">the</span> <span m="3934800">expected</span> <span m="3935370">value</span> <span m="3936050">of</span> <span m="3936190">c</span> <span m="3936460">to</span> <span m="3936550">the</span> <span m="3936650">T</span> <span m="3941230">is</span> <span m="3941520">at</span> <span m="3941650">most</span> <span m="3943390">a</span> <span m="3943510">product</span> <span m="3949216">of</span> <span m="3949800">expected</span> <span m="3950220">value</span> <span m="3950530">of</span> <span m="3950810">e</span> <span m="3951020">to</span> <span m="3951100">the</span> <span m="3951180">cTj</span> <span m="3952130">is</span> <span m="3952280">this--</span> <span m="3953570">e</span> <span m="3953860">to</span> <span m="3953910">the</span> <span m="3954030">c</span> <span m="3954370">minus</span> <span m="3954770">1</span> <span m="3955800">expected</span> <span m="3956280">value</span> <span m="3956760">of</span> <span m="3956850">Tj.</span> <span m="3959280">And</span> <span m="3959470">now</span> <span m="3959800">I</span> <span m="3959870">can</span> <span m="3960020">convert</span> <span m="3960330">this</span> <span m="3960490">to</span> <span m="3960620">a</span> <span m="3960690">sum</span> <span m="3961060">in</span> <span m="3961140">the</span> <span m="3961240">exponent.</span> <span m="3972630">And</span> <span m="3972760">this</span> <span m="3973706">is</span> <span m="3974180">j</span> <span m="3974300">equals</span> <span m="3974575">1</span> <span m="3974850">to n.</span> </p>
<p><span m="3976320">And</span> <span m="3976490">what</span> <span m="3976620">do</span> <span m="3976720">I</span> <span m="3976810">do</span> <span m="3976980">to</span> <span m="3977060">simplify</span> <span m="3977620">that?</span> <span m="3983760">Linearity</span> <span m="3984310">of</span> <span m="3984390">expectation.</span> <span m="3987500">c</span> <span m="3987770">minus</span> <span m="3988130">1</span> <span m="3989000">times</span> <span m="3989460">the</span> <span m="3989550">sum</span> <span m="3991070">j</span> <span m="3991320">equals</span> <span m="3991640">1</span> <span m="3991880">to</span> <span m="3992020">n</span> <span m="3993170">expected</span> <span m="3993720">value</span> <span m="3995130">of</span> <span m="3995430">Tj.</span> <span m="4001060">Ooh,</span> <span m="4001430">let's</span> <span m="4001740">see,</span> <span m="4001890">did</span> <span m="4002145">I?</span> <span m="4002400">Actually,</span> <span m="4002680">I</span> <span m="4002880">used</span> <span m="4003050">linearity</span> <span m="4004860">coming</span> <span m="4005200">out.</span> <span m="4005390">I</span> <span m="4005480">already</span> <span m="4005670">used</span> <span m="4005880">linearity.</span> <span m="4007510">I</span> <span m="4008000">screwed</span> <span m="4008310">up</span> <span m="4008440">here.</span> </p>
<p><span m="4010530">So</span> <span m="4010800">here</span> <span m="4011220">I</span> <span m="4011290">used</span> <span m="4011610">the</span> <span m="4011690">linearity</span> <span m="4012200">when</span> <span m="4012340">I</span> <span m="4012390">took</span> <span m="4012750">the</span> <span m="4012900">sum</span> <span m="4013400">up</span> <span m="4013580">here</span> <span m="4014520">inside</span> <span m="4015040">the</span> <span m="4015130">expectation.</span> <span m="4015700">I've</span> <span m="4015790">already used</span> <span m="4016100">linearity.</span> <span m="4016740">What</span> <span m="4017140">is</span> <span m="4017340">the</span> <span m="4017440">sum</span> <span m="4017790">of</span> <span m="4017870">the</span> <span m="4017940">Tj's?</span> <span m="4019810">T--</span> <span m="4020360">yeah,</span> <span m="4020540">that's</span> <span m="4020760">what</span> <span m="4020850">I</span> <span m="4020910">needed</span> <span m="4021040">to</span> <span m="4021170">do</span> <span m="4021360">here.</span> </p>
<p><span m="4027130">OK,</span> <span m="4028320">we're</span> <span m="4028480">now</span> <span m="4028730">almost</span> <span m="4029080">done.</span> <span m="4029360">We've</span> <span m="4029490">got</span> <span m="4029840">now</span> <span m="4030040">an</span> <span m="4030150">upper</span> <span m="4030400">bound</span> <span m="4030850">on</span> <span m="4030940">the</span> <span m="4031000">expected</span> <span m="4031430">value</span> <span m="4031650">of</span> <span m="4031710">c</span> <span m="4031870">to</span> <span m="4031950">the</span> <span m="4032040">T.</span> <span m="4032810">And</span> <span m="4032990">it</span> <span m="4033060">is</span> <span m="4033270">this.</span> <span m="4034450">And</span> <span m="4034590">we</span> <span m="4034680">just</span> <span m="4034900">plug</span> <span m="4035380">that</span> <span m="4035770">in</span> <span m="4036350">back</span> <span m="4036860">up</span> <span m="4036980">here.</span> <span m="4040750">So</span> <span m="4040860">now</span> <span m="4041050">this</span> <span m="4041310">is</span> <span m="4041410">at</span> <span m="4041560">most</span> <span m="4044270">e</span> <span m="4044580">to</span> <span m="4044680">the</span> <span m="4044770">c</span> <span m="4045050">minus</span> <span m="4045480">1</span> <span m="4046300">expected</span> <span m="4046860">value</span> <span m="4047260">of</span> <span m="4047370">T</span> <span m="4048870">over</span> <span m="4049565">c</span> <span m="4049840">to</span> <span m="4049920">the</span> <span m="4050010">c</span> <span m="4050360">times</span> <span m="4050670">the</span> <span m="4050750">expected</span> <span m="4051200">value</span> <span m="4051520">of</span> <span m="4051620">t.</span> <span m="4053810">And</span> <span m="4054010">now</span> <span m="4054210">I</span> <span m="4054280">just</span> <span m="4054500">do</span> <span m="4055020">manipulation.</span> <span m="4056390">c</span> <span m="4056840">to</span> <span m="4056890">something</span> <span m="4057400">is</span> <span m="4057550">the</span> <span m="4057620">same</span> <span m="4057870">as</span> <span m="4058020">e</span> <span m="4058380">to the</span> <span m="4058440">log</span> <span m="4058800">of</span> <span m="4058900">c</span> <span m="4059200">times</span> <span m="4059610">that</span> <span m="4059820">something.</span> <span m="4060770">So</span> <span m="4060970">this</span> <span m="4061250">is</span> <span m="4061610">e to</span> <span m="4061970">the</span> <span m="4063620">minus</span> <span m="4064200">c</span> <span m="4064688">ln</span> <span m="4065664">c</span> <span m="4066920">expected</span> <span m="4067350">value</span> <span m="4067630">of</span> <span m="4067690">T</span> <span m="4069530">plus</span> <span m="4070060">that.</span> </p>
<p><span m="4077220">And</span> <span m="4077460">then</span> <span m="4077780">I'm</span> <span m="4077920">running</span> <span m="4078180">out</span> <span m="4078610">of room.</span> <span m="4080760">That</span> <span m="4081190">equals--</span> <span m="4082790">I</span> <span m="4082920">can</span> <span m="4083090">just</span> <span m="4083480">pull</span> <span m="4083690">out</span> <span m="4083910">the</span> <span m="4084000">expected</span> <span m="4084440">values</span> <span m="4084860">of</span> <span m="4084960">T.</span> <span m="4085240">I</span> <span m="4085350">get</span> <span m="4085550">e</span> <span m="4085790">to</span> <span m="4085870">the</span> <span m="4085970">minus</span> <span m="4087270">c</span> <span m="4087690">log</span> <span m="4087980">of</span> <span m="4088060">c</span> <span m="4091380">plus</span> <span m="4091800">c</span> <span m="4092440">minus</span> <span m="4092830">1</span> <span m="4094120">expected</span> <span m="4094590">value</span> <span m="4094870">of</span> <span m="4094960">T.</span> <span m="4096819">And</span> <span m="4097080">that's</span> <span m="4097399">e to</span> <span m="4097689">the</span> <span m="4097750">minus</span> <span m="4098200">z</span> <span m="4099240">expected</span> <span m="4099689">value</span> <span m="4099920">of</span> <span m="4099990">T.</span> </p>
<p><span m="4103819">All</span> <span m="4104020">right, so</span> <span m="4104120">that's</span> <span m="4104319">a</span> <span m="4104359">marathon</span> <span m="4104950">proof.</span> <span m="4105290">It's</span> <span m="4105370">the</span> <span m="4105439">worst</span> <span m="4105740">proof</span> <span m="4106010">I</span> <span m="4106080">think.</span> <span m="4107370">Well,</span> <span m="4107510">maybe</span> <span m="4107720">minimum</span> <span m="4108020">spanning</span> <span m="4108260">tree</span> <span m="4108660">was</span> <span m="4108840">worse.</span> <span m="4108950">But this</span> <span m="4109149">is</span> <span m="4109330">one</span> <span m="4109490">of</span> <span m="4109550">the</span> <span m="4109609">worst</span> <span m="4109859">proofs</span> <span m="4110050">we've</span> <span m="4110170">seen</span> <span m="4110520">this</span> <span m="4110689">year.</span> <span m="4111340">But</span> <span m="4111540">I</span> <span m="4111590">wanted</span> <span m="4111979">to</span> <span m="4112420">show</span> <span m="4112770">it</span> <span m="4112850">to</span> <span m="4112990">you.</span> <span m="4114439">Because</span> <span m="4114729">it's</span> <span m="4114840">one</span> <span m="4114939">of</span> <span m="4115000">the</span> <span m="4115060">most</span> <span m="4115210">important</span> <span m="4115569">results</span> <span m="4116300">that</span> <span m="4116439">we</span> <span m="4116870">cover,</span> <span m="4117300">certainly</span> <span m="4117569">in</span> <span m="4117640">probability,</span> <span m="4119130">that</span> <span m="4119229">can</span> <span m="4119319">be</span> <span m="4119420">very</span> <span m="4119689">useful</span> <span m="4120279">in</span> <span m="4120420">practice.</span> <span m="4121460">And</span> <span m="4121580">it</span> <span m="4121640">gives</span> <span m="4121790">you</span> <span m="4121870">some</span> <span m="4122189">feel</span> <span m="4122450">for,</span> <span m="4122609">hey,</span> <span m="4122830">this</span> <span m="4123060">wasn't</span> <span m="4123350">so</span> <span m="4123500">obvious</span> <span m="4123810">to</span> <span m="4123899">do</span> <span m="4124010">it</span> <span m="4124090">the</span> <span m="4124160">first</span> <span m="4124420">time,</span> <span m="4125300">and</span> <span m="4125670">also</span> <span m="4125890">some</span> <span m="4126029">of</span> <span m="4126090">the</span> <span m="4126149">techniques</span> <span m="4126620">that</span> <span m="4126729">are</span> <span m="4126779">used,</span> <span m="4127640">which</span> <span m="4127859">is</span> <span m="4127979">really</span> <span m="4128180">Markov's</span> <span m="4128689">theorem.</span> <span m="4131210">Any</span> <span m="4131390">questions?</span> <span m="4133069">Yeah.</span> </p>
<p><span m="4134043">AUDIENCE: Over</span> <span m="4134530">there, you define</span> <span m="4135020">z</span> <span m="4135425">as</span> <span m="4135830">1</span> <span m="4136306">minus</span> <span m="4136782">c.</span> </p>
<p><span m="4138689">PROFESSOR: Did</span> <span m="4138790">I</span> <span m="4138830">do</span> <span m="4138970">it</span> <span m="4139140">wrong?</span> </p>
<p><span m="4139589">AUDIENCE: c</span> <span m="4140038">natural</span> <span m="4140487">log of c,</span> <span m="4140936">1 minus</span> <span m="4141385">c.</span> <span m="4141834">Maybe it's</span> <span m="4142283">plus c?</span> </p>
<p><span m="4143189">PROFESSOR: Oh,</span> <span m="4143460">I've</span> <span m="4143520">got</span> <span m="4143630">to</span> <span m="4143689">change</span> <span m="4143960">the</span> <span m="4144029">sign.</span> <span m="4144479">Because I</span> <span m="4144540">pulled</span> <span m="4144800">a</span> <span m="4144870">negative</span> <span m="4145330">out in</span> <span m="4145680">front.</span> <span m="4146399">So</span> <span m="4146510">it's</span> <span m="4146609">got</span> <span m="4146740">to</span> <span m="4146800">be</span> <span m="4146870">negative</span> <span m="4147399">c</span> <span m="4147640">minus</span> <span m="4148010">1,</span> <span m="4149180">which</span> <span m="4149380">means</span> <span m="4149670">negative</span> <span m="4149990">c</span> <span m="4150200">plus</span> <span m="4150520">1.</span> <span m="4151990">Yeah,</span> <span m="4152676">good.</span> <span m="4155600">Yeah,</span> <span m="4155790">this</span> <span m="4156010">was</span> <span m="4156160">OK.</span> <span m="4156510">I</span> <span m="4156590">just</span> <span m="4156779">made</span> <span m="4156930">the</span> <span m="4156960">mistake</span> <span m="4157310">going</span> <span m="4157520">to</span> <span m="4157590">there.</span> <span m="4158043">Any other</span> <span m="4158496">questions?</span> </p>
<p><span m="4164670">OK,</span> <span m="4165080">so</span> <span m="4168580">the</span> <span m="4169189">common</span> <span m="4169600">theme</span> <span m="4170000">here</span> <span m="4170229">in</span> <span m="4170340">using</span> <span m="4170640">Markov</span> <span m="4172790">to</span> <span m="4172899">get</span> <span m="4173120">Chebyshev,</span> <span m="4173779">to</span> <span m="4173850">get</span> <span m="4173990">Chernoff,</span> <span m="4174550">to</span> <span m="4174630">get</span> <span m="4174770">the</span> <span m="4174840">Markov</span> <span m="4175229">extensions,</span> <span m="4175840">is</span> <span m="4175990">always</span> <span m="4176380">the</span> <span m="4176479">same.</span> <span m="4177619">And</span> <span m="4178000">let</span> <span m="4178160">me</span> <span m="4178250">show</span> <span m="4178510">you</span> <span m="4178620">what</span> <span m="4178750">that</span> <span m="4179224">theme</span> <span m="4180172">is.</span> <span m="4186300">Because</span> <span m="4186510">you</span> <span m="4186620">can</span> <span m="4186760">use</span> <span m="4186960">it</span> <span m="4187069">to</span> <span m="4187149">get</span> <span m="4187319">even</span> <span m="4187609">other</span> <span m="4189340">results.</span> </p>
<p><span m="4192569">When</span> <span m="4192700">we're</span> <span m="4192790">trying</span> <span m="4193050">to</span> <span m="4193120">figure</span> <span m="4193380">out</span> <span m="4193529">the</span> <span m="4193609">probability</span> <span m="4194109">that</span> <span m="4194270">T</span> <span m="4194590">is</span> <span m="4194730">at</span> <span m="4194840">least</span> <span m="4195170">c</span> <span m="4195600">times</span> <span m="4196930">its</span> <span m="4197120">expected</span> <span m="4197640">value,</span> <span m="4200570">or</span> <span m="4201020">actually</span> <span m="4201380">even</span> <span m="4201540">in</span> <span m="4201630">general,</span> <span m="4201980">even</span> <span m="4202330">more</span> <span m="4202640">generally</span> <span m="4203000">than</span> <span m="4203160">that,</span> <span m="4204570">the</span> <span m="4204700">probability</span> <span m="4205760">that</span> <span m="4205940">A</span> <span m="4206300">is</span> <span m="4206480">bigger</span> <span m="4206740">than</span> <span m="4206930">B,</span> <span m="4208030">even</span> <span m="4208250">more</span> <span m="4208450">generally,</span> <span m="4210460">well,</span> <span m="4210620">that's</span> <span m="4210800">the</span> <span m="4210890">same</span> <span m="4211250">as</span> <span m="4211360">the</span> <span m="4211430">probability</span> <span m="4211910">that</span> <span m="4212100">f</span> <span m="4212330">of</span> <span m="4212470">A</span> <span m="4213750">is</span> <span m="4213970">bigger</span> <span m="4214240">than</span> <span m="4214500">f</span> <span m="4214730">of</span> <span m="4214850">B</span> <span m="4216150">as</span> <span m="4216260">long</span> <span m="4216440">as</span> <span m="4216540">you</span> <span m="4216600">don't</span> <span m="4216760">change</span> <span m="4217100">signs.</span> <span m="4220010">And</span> <span m="4220110">then</span> <span m="4220210">by</span> <span m="4220330">Markov,</span> <span m="4220940">this</span> <span m="4221150">is</span> <span m="4221250">at</span> <span m="4221350">most</span> <span m="4222450">the</span> <span m="4222550">expected</span> <span m="4222960">value</span> <span m="4223310">of</span> <span m="4223400">that</span> <span m="4225100">as</span> <span m="4225180">long</span> <span m="4225360">as</span> <span m="4225430">it's</span> <span m="4225570">non-negative</span> <span m="4227480">over</span> <span m="4227760">that.</span> </p>
<p><span m="4231600">In</span> <span m="4231850">Chebyshev,</span> <span m="4232990">what</span> <span m="4233320">function</span> <span m="4233740">f</span> <span m="4233970">did</span> <span m="4234120">we</span> <span m="4234250">use</span> <span m="4234580">for</span> <span m="4234720">Chebyshev</span> <span m="4235215">in</span> <span m="4235710">deriving</span> <span m="4236260">Chebyshev's</span> <span m="4236720">theorem?</span> <span m="4239540">What</span> <span m="4239710">was</span> <span m="4239900">f</span> <span m="4240170">doing</span> <span m="4240500">in</span> <span m="4240610">Chebyshev?</span> <span m="4241260">Actually</span> <span m="4241530">I</span> <span m="4241870">probably</span> <span m="4242020">just</span> <span m="4242380">erased</span> <span m="4242817">it.</span> <span m="4246120">What</span> <span m="4246460">operation</span> <span m="4246830">were we</span> <span m="4247200">doing with</span> <span m="4247360">Chebyshev?</span> </p>
<p><span m="4247876">AUDIENCE: Variance.</span> </p>
<p><span m="4249620">PROFESSOR: Variance.</span> <span m="4250520">And</span> <span m="4250610">that</span> <span m="4250800">meant</span> <span m="4251030">we</span> <span m="4251150">were</span> <span m="4251460">squaring</span> <span m="4252310">it.</span> <span m="4254500">So</span> <span m="4254780">the</span> <span m="4255340">technique</span> <span m="4255800">used</span> <span m="4256010">to</span> <span m="4256070">prove</span> <span m="4256240">Chebyshev</span> <span m="4257000">was</span> <span m="4257960">f</span> <span m="4258190">was</span> <span m="4258340">the</span> <span m="4258420">square</span> <span m="4258750">function.</span> <span m="4259650">For</span> <span m="4259790">Chernoff,</span> <span m="4261200">f</span> <span m="4261490">is</span> <span m="4261670">the</span> <span m="4261790">exponentiation</span> <span m="4262660">function,</span> <span m="4264380">which</span> <span m="4264570">turns</span> <span m="4264860">out</span> <span m="4265060">to</span> <span m="4265160">be--</span> <span m="4265730">in</span> <span m="4265860">fact,</span> <span m="4266080">when we</span> <span m="4266220">did</span> <span m="4266340">it</span> <span m="4266400">for</span> <span m="4266490">Chernoff,</span> <span m="4267560">that's</span> <span m="4267810">the</span> <span m="4267950">optimal</span> <span m="4268740">choice</span> <span m="4269100">of</span> <span m="4269190">functions</span> <span m="4269647">to get good</span> <span m="4270104">bounds.</span> </p>
<p><span m="4273776">All</span> <span m="4274240">right,</span> <span m="4274450">any</span> <span m="4274590">questions</span> <span m="4276800">on</span> <span m="4276990">that?</span> <span m="4278301">All right,</span> <span m="4278740">let's</span> <span m="4279010">do one</span> <span m="4279240">more</span> <span m="4279400">example</span> <span m="4281030">here</span> <span m="4281500">with</span> <span m="4281976">numbers.</span> <span m="4286740">And</span> <span m="4286870">this</span> <span m="4287050">is</span> <span m="4287180">a</span> <span m="4287390">load</span> <span m="4287810">balancing</span> <span m="4288400">application</span> <span m="4289480">for</span> <span m="4289490">example</span> <span m="4289940">you</span> <span m="4290040">might</span> <span m="4290100">have</span> <span m="4290360">with</span> <span m="4290510">web</span> <span m="4290750">servers.</span> <span m="4293450">Say</span> <span m="4293650">you've</span> <span m="4293790">got</span> <span m="4293890">to</span> <span m="4293950">build</span> <span m="4294120">a</span> <span m="4294150">load</span> <span m="4294470">balancing</span> <span m="4294890">device,</span> <span m="4296080">and</span> <span m="4296210">it's</span> <span m="4296310">got</span> <span m="4296450">to</span> <span m="4296520">balance</span> <span m="4297400">N</span> <span m="4297670">jobs,</span> <span m="4299570">B1,</span> <span m="4301250">B2,</span> <span m="4302480">to</span> <span m="4302930">BN,</span> <span m="4304880">across</span> <span m="4305420">a</span> <span m="4305480">set</span> <span m="4305770">of</span> <span m="4305900">M</span> <span m="4306180">servers,</span> <span m="4310440">S1,</span> <span m="4311530">S2,</span> <span m="4313200">to</span> <span m="4313420">SN.</span> </p>
<p><span m="4315910">And</span> <span m="4316530">say</span> <span m="4316660">you're</span> <span m="4316850">doing</span> <span m="4317100">this</span> <span m="4317270">for</span> <span m="4317460">a</span> <span m="4317740">decent</span> <span m="4318130">sized</span> <span m="4318480">website.</span> <span m="4319600">So</span> <span m="4320000">maybe</span> <span m="4320550">N</span> <span m="4320980">is</span> <span m="4321720">100,000.</span> <span m="4323290">You</span> <span m="4323370">get</span> <span m="4323460">100,000</span> <span m="4324060">requests</span> <span m="4324440">a</span> <span m="4324490">minute.</span> <span m="4326060">And</span> <span m="4326340">say</span> <span m="4326770">you've</span> <span m="4326890">got</span> <span m="4327360">10</span> <span m="4327640">servers</span> <span m="4329330">to</span> <span m="4329490">handle</span> <span m="4329810">those</span> <span m="4330030">requests.</span> <span m="4331700">And</span> <span m="4332250">say</span> <span m="4332530">the</span> <span m="4332680">requests</span> <span m="4334190">are--</span> <span m="4335070">the</span> <span m="4335370">time</span> <span m="4335710">for</span> <span m="4335800">the</span> <span m="4335910">j-th</span> <span m="4336170">request</span> <span m="4336740">is,</span> <span m="4338220">say,</span> <span m="4338360">Bj</span> <span m="4339830">takes</span> <span m="4340250">the</span> <span m="4340420">j-th</span> <span m="4340720">job.</span> <span m="4341130">The j-th</span> <span m="4341440">request</span> <span m="4341920">takes</span> <span m="4342890">Lj</span> <span m="4343610">time.</span> <span m="4345600">And</span> <span m="4345740">the</span> <span m="4345810">time</span> <span m="4346200">is</span> <span m="4346330">the</span> <span m="4346410">same</span> <span m="4346650">on</span> <span m="4346770">any</span> <span m="4346930">server.</span> <span m="4347330">The</span> <span m="4347420">servers</span> <span m="4347820">are</span> <span m="4347900">all</span> <span m="4348030">equivalent.</span> </p>
<p><span m="4349480">And</span> <span m="4349720">let's</span> <span m="4349900">assume</span> <span m="4350180">it's</span> <span m="4350300">normalized</span> <span m="4351230">so</span> <span m="4351330">that</span> <span m="4351750">Lj</span> <span m="4352370">is</span> <span m="4352500">between</span> <span m="4353010">0</span> <span m="4353320">and</span> <span m="4353460">1.</span> <span m="4354850">Maybe</span> <span m="4355520">the</span> <span m="4355850">worst</span> <span m="4356240">job</span> <span m="4356540">takes</span> <span m="4356800">a</span> <span m="4356860">second</span> <span m="4357210">to</span> <span m="4357300">do,</span> <span m="4357780">let's</span> <span m="4357990">say.</span> <span m="4360490">And</span> <span m="4360910">say</span> <span m="4361190">that</span> <span m="4361290">if</span> <span m="4361360">you</span> <span m="4361490">sum</span> <span m="4361900">up</span> <span m="4362240">the</span> <span m="4362320">length</span> <span m="4362650">of</span> <span m="4362760">all</span> <span m="4363120">the</span> <span m="4363200">jobs,</span> <span m="4364050">you</span> <span m="4364190">get</span> <span m="4364790">L.</span> <span m="4366210">Total</span> <span m="4366580">workload</span> <span m="4370060">is</span> <span m="4370140">the</span> <span m="4370230">sum</span> <span m="4370510">of</span> <span m="4370620">all</span> <span m="4370830">of</span> <span m="4370920">them.</span> <span m="4373780">j equals</span> <span m="4374220">1</span> <span m="4374720">to</span> <span m="4375203">N.</span> </p>
<p><span m="4378180">And</span> <span m="4378280">we're</span> <span m="4378420">going</span> <span m="4378550">to</span> <span m="4378630">assume</span> <span m="4379340">that</span> <span m="4379450">the</span> <span m="4379670">average</span> <span m="4380400">job</span> <span m="4380760">length</span> <span m="4381000">is</span> <span m="4381100">1/4</span> <span m="4381480">second.</span> <span m="4382890">So</span> <span m="4383260">we're</span> <span m="4383360">going</span> <span m="4383480">to</span> <span m="4383540">assume</span> <span m="4384150">that</span> <span m="4386760">the</span> <span m="4386860">total</span> <span m="4388170">amount</span> <span m="4388450">of</span> <span m="4388520">work</span> <span m="4388820">is</span> <span m="4388920">25,000</span> <span m="4391780">seconds,</span> <span m="4392250">say.</span> <span m="4392810">So</span> <span m="4392960">the</span> <span m="4393080">average</span> <span m="4393500">job</span> <span m="4393760">length</span> <span m="4393970">is</span> <span m="4394070">1/4</span> <span m="4394550">second.</span> <span m="4396580">And</span> <span m="4396800">the</span> <span m="4396860">job</span> <span m="4397430">is</span> <span m="4397600">to</span> <span m="4397720">assign</span> <span m="4398170">these</span> <span m="4398370">tasks</span> <span m="4398760">to</span> <span m="4398820">the</span> <span m="4398890">10</span> <span m="4399180">servers</span> <span m="4400920">so</span> <span m="4401060">that</span> <span m="4401410">hopefully</span> <span m="4402260">every</span> <span m="4402600">server</span> <span m="4403120">is</span> <span m="4403270">doing</span> <span m="4404860">L/M</span> <span m="4405520">work,</span> <span m="4405970">which</span> <span m="4406140">would</span> <span m="4406230">be</span> <span m="4406320">25,000/10,</span> <span m="4410000">or</span> <span m="4410240">2,500</span> <span m="4411770">milliseconds</span> <span m="4412500">of</span> <span m="4412580">work,</span> <span m="4413360">something</span> <span m="4413630">like</span> <span m="4413830">that.</span> <span m="4414300">I</span> <span m="4414320">don't</span> <span m="4414460">know.</span> </p>
<p><span m="4416330">Because</span> <span m="4416510">when</span> <span m="4416630">you're</span> <span m="4416720">doing</span> <span m="4416870">load</span> <span m="4417200">balancing,</span> <span m="4417740">you</span> <span m="4417800">want</span> <span m="4417920">to</span> <span m="4417980">take</span> <span m="4418190">your</span> <span m="4418300">load</span> <span m="4418550">and</span> <span m="4418660">spread</span> <span m="4418920">it</span> <span m="4419050">evenly</span> <span m="4419570">and</span> <span m="4419810">equally</span> <span m="4420130">among</span> <span m="4420380">all</span> <span m="4420540">the</span> <span m="4420620">servers.</span> <span m="4423540">Any</span> <span m="4423700">questions</span> <span m="4424130">about</span> <span m="4424330">the</span> <span m="4424400">problem?</span> <span m="4426720">You've</span> <span m="4426880">got</span> <span m="4427450">a</span> <span m="4427480">bunch</span> <span m="4427670">of</span> <span m="4427740">jobs,</span> <span m="4428120">a</span> <span m="4428150">bunch</span> <span m="4428330">of</span> <span m="4428400">servers.</span> <span m="4428800">You</span> <span m="4428870">want</span> <span m="4428990">to</span> <span m="4429050">assign</span> <span m="4429350">the</span> <span m="4429410">jobs</span> <span m="4429750">to</span> <span m="4429810">the</span> <span m="4429870">servers</span> <span m="4430300">to</span> <span m="4430830">balance</span> <span m="4431290">the</span> <span m="4431370">load.</span> <span m="4433570">Well,</span> <span m="4435220">what</span> <span m="4435610">is</span> <span m="4435770">the</span> <span m="4435930">simplest</span> <span m="4436390">algorithm</span> <span m="4436720">you</span> <span m="4436810">could</span> <span m="4436920">think</span> <span m="4437160">of</span> <span m="4437250">to</span> <span m="4437340">do</span> <span m="4437500">this?</span> </p>
<p><span m="4438410">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4440230">PROFESSOR: That's</span> <span m="4440440">a</span> <span m="4440480">good</span> <span m="4440760">algorithm</span> <span m="4441140">to</span> <span m="4441240">do</span> <span m="4441420">this.</span> <span m="4442050">In</span> <span m="4442240">practice,</span> <span m="4443650">the</span> <span m="4443770">first</span> <span m="4444080">thing</span> <span m="4444260">people</span> <span m="4444540">do</span> <span m="4444820">is,</span> <span m="4444930">well,</span> <span m="4445560">take</span> <span m="4445800">the</span> <span m="4445870">first</span> <span m="4449320">N/M</span> <span m="4449870">jobs,</span> <span m="4450260">put</span> <span m="4450390">them on</span> <span m="4450550">server</span> <span m="4450860">one,</span> <span m="4451910">the</span> <span m="4452040">next</span> <span m="4452370">N/M</span> <span m="4452850">on</span> <span m="4452950">server</span> <span m="4453270">two.</span> <span m="4454560">Or</span> <span m="4455070">they'll</span> <span m="4455200">use</span> <span m="4455420">something</span> <span m="4455630">called</span> <span m="4455890">round</span> <span m="4456300">robin--</span> <span m="4457250">first</span> <span m="4457530">job</span> <span m="4457750">goes</span> <span m="4457970">here,</span> <span m="4458220">second</span> <span m="4458610">here,</span> <span m="4458910">third</span> <span m="4459280">here,</span> <span m="4459570">10th</span> <span m="4460010">here,</span> <span m="4460490">back</span> <span m="4460760">and</span> <span m="4460850">start</span> <span m="4461100">over.</span> <span m="4462460">And</span> <span m="4462670">they</span> <span m="4462790">hope</span> <span m="4464070">that</span> <span m="4464230">it</span> <span m="4464290">will</span> <span m="4464380">balance</span> <span m="4464790">the</span> <span m="4464860">load.</span> <span m="4465230">But</span> <span m="4465560">it</span> <span m="4465960">might</span> <span m="4466230">well</span> <span m="4466420">not.</span> <span m="4467340">Because</span> <span m="4467780">maybe</span> <span m="4468130">every</span> <span m="4468350">10th</span> <span m="4468670">job</span> <span m="4468980">is</span> <span m="4469070">a</span> <span m="4469130">big</span> <span m="4469320">one.</span> </p>
<p><span m="4471500">So</span> <span m="4471600">what's</span> <span m="4471780">much</span> <span m="4472140">better</span> <span m="4472320">to</span> <span m="4472390">do</span> <span m="4472530">in</span> <span m="4472610">practice</span> <span m="4473110">is</span> <span m="4473250">to</span> <span m="4473360">assign</span> <span m="4473810">them</span> <span m="4474000">randomly.</span> <span m="4475370">So</span> <span m="4475560">a</span> <span m="4475600">job</span> <span m="4475990">comes</span> <span m="4476290">in.</span> <span m="4477370">You</span> <span m="4477470">don't</span> <span m="4477630">even</span> <span m="4478020">pay</span> <span m="4478230">attention</span> <span m="4478670">to</span> <span m="4478750">how</span> <span m="4478950">hard</span> <span m="4479290">it</span> <span m="4479390">is,</span> <span m="4480220">how</span> <span m="4480400">much</span> <span m="4480560">time</span> <span m="4480910">you</span> <span m="4481010">think</span> <span m="4481090">it'll</span> <span m="4481230">take.</span> <span m="4481510">You</span> <span m="4481600">might</span> <span m="4481790">not</span> <span m="4481940">even</span> <span m="4482120">know</span> <span m="4483160">before</span> <span m="4483460">you</span> <span m="4483540">start</span> <span m="4483790">the</span> <span m="4483860">job</span> <span m="4484130">how</span> <span m="4484280">long</span> <span m="4484440">it's</span> <span m="4484530">going</span> <span m="4484650">to</span> <span m="4484710">take</span> <span m="4484870">to</span> <span m="4484940">complete.</span> <span m="4485920">Give</span> <span m="4486140">it</span> <span m="4486220">to</span> <span m="4486340">a</span> <span m="4486440">random</span> <span m="4486780">server.</span> <span m="4488280">Don't</span> <span m="4488500">even</span> <span m="4488670">look</span> <span m="4488980">at</span> <span m="4489060">how</span> <span m="4489200">much</span> <span m="4489390">work</span> <span m="4489590">that</span> <span m="4489730">server</span> <span m="4490030">has.</span> <span m="4490990">Just</span> <span m="4491290">give</span> <span m="4491430">it</span> <span m="4491500">to</span> <span m="4491620">a</span> <span m="4491690">random</span> <span m="4492030">one.</span> </p>
<p><span m="4493550">And</span> <span m="4493660">it</span> <span m="4493720">turns</span> <span m="4494050">out</span> <span m="4494350">this</span> <span m="4494510">does</span> <span m="4494730">very,</span> <span m="4495380">very</span> <span m="4495580">well.</span> <span m="4496550">Without</span> <span m="4496870">knowing</span> <span m="4497150">anything,</span> <span m="4497610">just</span> <span m="4497850">that</span> <span m="4498110">simple</span> <span m="4498450">approach</span> <span m="4498950">does</span> <span m="4499190">great</span> <span m="4499940">in</span> <span m="4500080">practice.</span> <span m="4500940">And</span> <span m="4501320">today,</span> <span m="4501780">state</span> <span m="4502100">of</span> <span m="4502180">the</span> <span m="4502280">art</span> <span m="4503700">load</span> <span m="4504030">balancers</span> <span m="4504485">do</span> <span m="4504940">this.</span> <span m="4508610">We've</span> <span m="4508750">been</span> <span m="4508880">doing</span> <span m="4509640">randomized</span> <span m="4510290">kinds</span> <span m="4510570">of</span> <span m="4510640">thing</span> <span m="4510780">like</span> <span m="4510960">this</span> <span m="4511120">at</span> <span m="4511240">Akamai</span> <span m="4511710">now</span> <span m="4512160">for</span> <span m="4512290">a</span> <span m="4512340">decade.</span> <span m="4512780">And</span> <span m="4512870">it's</span> <span m="4512990">just</span> <span m="4513270">stunning</span> <span m="4514880">how</span> <span m="4515190">well</span> <span m="4515420">it</span> <span m="4515510">works.</span> <span m="4516820">And</span> <span m="4516880">so</span> <span m="4516980">let's</span> <span m="4517150">see</span> <span m="4518060">why</span> <span m="4518330">that</span> <span m="4518500">is.</span> </p>
<p><span m="4524730">Of</span> <span m="4524820">course</span> <span m="4525070">we're</span> <span m="4525140">going</span> <span m="4525260">to</span> <span m="4525320">use</span> <span m="4525590">the</span> <span m="4525670">Chernoff</span> <span m="4526130">bound</span> <span m="4526390">to</span> <span m="4526450">do</span> <span m="4526660">it.</span> <span m="4528320">So</span> <span m="4528530">let's</span> <span m="4528830">let</span> <span m="4531140">Rij</span> <span m="4532952">be</span> <span m="4533290">the</span> <span m="4533570">load</span> <span m="4536130">on</span> <span m="4538730">server</span> <span m="4540250">Si</span> <span m="4541840">from</span> <span m="4543390">job</span> <span m="4544840">Bj.</span> <span m="4547080">Now,</span> <span m="4548840">if</span> <span m="4550030">Bj</span> <span m="4550700">is</span> <span m="4550820">not</span> <span m="4551020">assigned</span> <span m="4551670">to</span> <span m="4551750">Si,</span> <span m="4553220">it's</span> <span m="4553400">zero</span> <span m="4553760">load.</span> <span m="4554470">Because</span> <span m="4554650">it's</span> <span m="4554750">not</span> <span m="4554950">even</span> <span m="4555260">doing</span> <span m="4555470">the</span> <span m="4555630">work</span> <span m="4555890">there.</span> <span m="4557040">So</span> <span m="4557290">we</span> <span m="4557380">know</span> <span m="4557550">that</span> <span m="4557690">Rij</span> <span m="4558740">equals</span> <span m="4561520">the</span> <span m="4561650">load</span> <span m="4562100">of</span> <span m="4562520">Bj</span> <span m="4563950">if</span> <span m="4564190">it's</span> <span m="4564330">assigned</span> <span m="4566370">to</span> <span m="4566500">Si.</span> <span m="4571020">And</span> <span m="4571200">that</span> <span m="4571390">happens</span> <span m="4571800">with</span> <span m="4571910">probability</span> <span m="4573470">1/M.</span> <span m="4575890">The</span> <span m="4575980">job</span> <span m="4577290">picks</span> <span m="4577540">one</span> <span m="4577790">of</span> <span m="4577860">the</span> <span m="4577950">M</span> <span m="4578040">servers</span> <span m="4578430">at</span> <span m="4578540">random.</span> <span m="4579610">And</span> <span m="4579810">otherwise,</span> <span m="4580470">the</span> <span m="4580530">load</span> <span m="4580730">is</span> <span m="4580820">0.</span> <span m="4582750">Because</span> <span m="4582910">it's</span> <span m="4583050">not</span> <span m="4583240">assigned</span> <span m="4583560">to</span> <span m="4583620">that</span> <span m="4583800">server.</span> <span m="4585770">And</span> <span m="4585840">that</span> <span m="4586040">is</span> <span m="4586210">probability</span> <span m="4587790">1</span> <span m="4588160">minus</span> <span m="4588490">1/M.</span> </p>
<p><span m="4591860">Now</span> <span m="4592170">let's</span> <span m="4592430">look</span> <span m="4592610">at</span> <span m="4592700">how</span> <span m="4592920">much</span> <span m="4593190">load</span> <span m="4593660">gets</span> <span m="4593850">assigned</span> <span m="4594370">by</span> <span m="4594490">this</span> <span m="4594700">random</span> <span m="4595130">algorithm</span> <span m="4596490">to</span> <span m="4596610">server</span> <span m="4597040">i.</span> <span m="4599140">So we'll</span> <span m="4599560">let</span> <span m="4599930">Ri</span> <span m="4602440">be</span> <span m="4605030">the</span> <span m="4605140">sum</span> <span m="4605510">of</span> <span m="4605610">all</span> <span m="4605850">the</span> <span m="4605920">load</span> <span m="4608100">assigned</span> <span m="4608890">to</span> <span m="4609000">server</span> <span m="4609370">i.</span> <span m="4618830">So</span> <span m="4618970">we've</span> <span m="4619120">got</span> <span m="4619470">this</span> <span m="4621130">indicator</span> <span m="4621700">where</span> <span m="4621830">the</span> <span m="4622010">random</span> <span m="4622370">variables</span> <span m="4622710">are</span> <span m="4622740">not</span> <span m="4622880">0, 1.</span> <span m="4623510">They're</span> <span m="4623650">0</span> <span m="4624100">and</span> <span m="4624230">whatever</span> <span m="4624500">this</span> <span m="4624900">load</span> <span m="4625240">happens</span> <span m="4625560">to</span> <span m="4625630">be</span> <span m="4625760">for</span> <span m="4625860">the</span> <span m="4625930">j-th</span> <span m="4626220">job,</span> <span m="4626610">at</span> <span m="4626720">most</span> <span m="4627030">1.</span> <span m="4628540">And</span> <span m="4628980">we</span> <span m="4629080">sum</span> <span m="4629340">up</span> <span m="4629850">the</span> <span m="4629990">value</span> <span m="4630390">for</span> <span m="4631370">the</span> <span m="4631510">contribution</span> <span m="4632180">to</span> <span m="4632260">Si</span> <span m="4633390">over all</span> <span m="4633860">the</span> <span m="4633930">jobs.</span> </p>
<p><span m="4637360">So</span> <span m="4637390">now</span> <span m="4637620">we</span> <span m="4637710">compute</span> <span m="4638090">the</span> <span m="4638180">expected</span> <span m="4638800">value</span> <span m="4639500">of</span> <span m="4639660">Ri,</span> <span m="4640363">the</span> <span m="4640816">expected load on</span> <span m="4641269">the</span> <span m="4641722">i-th</span> <span m="4642175">server.</span> <span m="4648500">So</span> <span m="4648820">the</span> <span m="4648900">expected</span> <span m="4649760">load</span> <span m="4650110">on</span> <span m="4650230">the</span> <span m="4650320">i-th</span> <span m="4650580">server</span> <span m="4652930">is--</span> <span m="4654215">well,</span> <span m="4654570">we</span> <span m="4654870">use</span> <span m="4655100">linearity</span> <span m="4655440">of</span> <span m="4655530">expectation.</span> <span m="4664870">And</span> <span m="4665280">the</span> <span m="4665360">expected</span> <span m="4665930">value</span> <span m="4666240">of</span> <span m="4666330">Rij--</span> <span m="4668650">well,</span> <span m="4668980">0</span> <span m="4669450">or</span> <span m="4669550">Lj.</span> <span m="4670070">It's</span> <span m="4670520">Lj</span> <span m="4670930">with</span> <span m="4671070">probability</span> <span m="4671470">1/M.</span> <span m="4673420">This</span> <span m="4673650">is</span> <span m="4673770">just</span> <span m="4674020">now</span> <span m="4674250">the</span> <span m="4674350">sum</span> <span m="4676545">of</span> <span m="4676820">Lj</span> <span m="4677155">over</span> <span m="4677490">M.</span> <span m="4680270">And</span> <span m="4680470">the</span> <span m="4680540">sum</span> <span m="4680950">of</span> <span m="4681090">Lj</span> <span m="4681760">is</span> <span m="4681880">just</span> <span m="4682200">L.</span> </p>
<p><span m="4686060">So</span> <span m="4686300">the</span> <span m="4686420">expected</span> <span m="4687070">load</span> <span m="4688030">of</span> <span m="4688240">the</span> <span m="4688380">i-th</span> <span m="4688570">server</span> <span m="4690050">is</span> <span m="4690230">the</span> <span m="4690310">total</span> <span m="4690710">load</span> <span m="4690980">divided</span> <span m="4691330">by</span> <span m="4691420">the</span> <span m="4691500">number</span> <span m="4691740">of</span> <span m="4691810">servers,</span> <span m="4692630">which</span> <span m="4692880">is</span> <span m="4693080">perfect.</span> <span m="4694210">It's</span> <span m="4694310">optimal--</span> <span m="4695780">can't</span> <span m="4696120">do</span> <span m="4696230">better</span> <span m="4696450">than</span> <span m="4696570">that.</span> <span m="4700380">It</span> <span m="4700510">makes</span> <span m="4700740">sense.</span> <span m="4700970">If</span> <span m="4701030">you</span> <span m="4701120">assign</span> <span m="4701550">all</span> <span m="4701760">the</span> <span m="4701830">jobs</span> <span m="4702240">randomly,</span> <span m="4703600">every</span> <span m="4703860">server</span> <span m="4704320">is</span> <span m="4704420">expecting</span> <span m="4704940">to</span> <span m="4705010">get</span> <span m="4705250">1/M</span> <span m="4706050">of</span> <span m="4706140">the</span> <span m="4706210">total</span> <span m="4706530">load.</span> </p>
<p><span m="4709530">Now</span> <span m="4709770">we</span> <span m="4709870">want</span> <span m="4710040">to</span> <span m="4710100">know</span> <span m="4710210">the</span> <span m="4710300">probability</span> <span m="4710860">it</span> <span m="4710950">deviates</span> <span m="4711490">from</span> <span m="4711660">that,</span> <span m="4712412">that</span> <span m="4712790">you</span> <span m="4712890">have</span> <span m="4713100">too</span> <span m="4713260">much</span> <span m="4713550">load</span> <span m="4713790">on</span> <span m="4713890">the</span> <span m="4714000">i-th</span> <span m="4714498">server.</span> <span m="4731460">All</span> <span m="4731580">right,</span> <span m="4731790">so</span> <span m="4731890">the</span> <span m="4731970">probability</span> <span m="4733540">that</span> <span m="4733840">the</span> <span m="4734000">i-th</span> <span m="4734320">server</span> <span m="4735200">has</span> <span m="4735760">c</span> <span m="4736430">times</span> <span m="4737040">the</span> <span m="4737180">optimal</span> <span m="4737620">load</span> <span m="4740870">is</span> <span m="4741110">at</span> <span m="4741210">most,</span> <span m="4741770">by</span> <span m="4741920">Chernoff,</span> <span m="4742530">if</span> <span m="4742650">the</span> <span m="4742930">jobs</span> <span m="4743340">are</span> <span m="4743400">independent,</span> <span m="4744830">minus</span> <span m="4745240">zL</span> <span m="4745657">over</span> <span m="4746074">M,</span> <span m="4746910">minus</span> <span m="4747610">z</span> <span m="4748140">times</span> <span m="4748560">the</span> <span m="4748660">expected</span> <span m="4749230">load</span> <span m="4750540">where</span> <span m="4750750">z</span> <span m="4751340">is</span> <span m="4752410">c</span> <span m="4752780">ln</span> <span m="4753340">c</span> <span m="4755050">plus</span> <span m="4755400">1</span> <span m="4755630">minus</span> <span m="4755930">c.</span> <span m="4756530">This</span> <span m="4756730">is</span> <span m="4756850">Chernoff</span> <span m="4757380">now,</span> <span m="4758280">just</span> <span m="4758540">straight</span> <span m="4758780">from</span> <span m="4758880">the</span> <span m="4758940">formula</span> <span m="4759370">of</span> <span m="4759600">Chernoff,</span> <span m="4759770">as</span> <span m="4759890">long</span> <span m="4760050">as</span> <span m="4760150">these</span> <span m="4760380">loads</span> <span m="4760650">are</span> <span m="4760700">mutually</span> <span m="4760870">independent.</span> </p>
<p><span m="4763110">All</span> <span m="4763350">right,</span> <span m="4763650">so</span> <span m="4767420">we</span> <span m="4767530">know</span> <span m="4767800">that</span> <span m="4767930">when</span> <span m="4768040">c</span> <span m="4768350">gets</span> <span m="4768550">to</span> <span m="4768650">be--</span> <span m="4768970">I</span> <span m="4769070">don't</span> <span m="4769200">know,</span> <span m="4769320">you</span> <span m="4769430">pick</span> <span m="4770460">10%</span> <span m="4771620">above</span> <span m="4771860">optimal,</span> <span m="4773450">c</span> <span m="4774070">equals</span> <span m="4774530">1.1,</span> <span m="4776970">well,</span> <span m="4777730">we</span> <span m="4777910">know</span> <span m="4778210">that</span> <span m="4778370">this</span> <span m="4778550">is</span> <span m="4778640">going</span> <span m="4778760">to</span> <span m="4778820">be</span> <span m="4778920">a</span> <span m="4778980">very</span> <span m="4779570">small</span> <span m="4779970">number.</span> <span m="4780860">L/M</span> <span m="4781580">is</span> <span m="4781680">2,500.</span> <span m="4782670">And</span> <span m="4783030">z,</span> <span m="4783430">in</span> <span m="4783530">this</span> <span m="4783710">case,</span> <span m="4784150">we</span> <span m="4784270">found</span> <span m="4784700">was</span> <span m="4785810">0.0048.</span> <span m="4788730">So</span> <span m="4788890">we</span> <span m="4788970">get</span> <span m="4789420">e</span> <span m="4789460">to</span> <span m="4789500">the</span> <span m="4789610">minus</span> <span m="4791920">0.0048</span> <span m="4794200">times</span> <span m="4794700">2,500.</span> <span m="4796970">And</span> <span m="4797160">that</span> <span m="4797400">is</span> <span m="4797630">really</span> <span m="4797910">tiny.</span> <span m="4798400">That's</span> <span m="4799610">less</span> <span m="4799890">than</span> <span m="4800030">1</span> <span m="4800300">in</span> <span m="4800390">160,000.</span> </p>
<p><span m="4804730">So</span> <span m="4805440">Chernoff</span> <span m="4805840">tells</span> <span m="4806180">us</span> <span m="4806300">the</span> <span m="4806380">probability</span> <span m="4806960">that</span> <span m="4807100">any</span> <span m="4807350">server,</span> <span m="4808140">a</span> <span m="4808190">particular</span> <span m="4808520">server,</span> <span m="4808840">gets</span> <span m="4809080">10%</span> <span m="4809850">load</span> <span m="4810440">more</span> <span m="4810680">than</span> <span m="4810810">you</span> <span m="4810910">expect</span> <span m="4812190">is</span> <span m="4812380">minuscule.</span> <span m="4815220">Now,</span> <span m="4815350">we're</span> <span m="4815440">not</span> <span m="4815800">quite</span> <span m="4816170">done.</span> <span m="4818230">That</span> <span m="4818460">tells</span> <span m="4818710">us</span> <span m="4818830">the</span> <span m="4818890">probability</span> <span m="4819410">the</span> <span m="4819490">first</span> <span m="4819890">server</span> <span m="4820710">gets</span> <span m="4821150">10%</span> <span m="4821690">too</span> <span m="4821800">much</span> <span m="4822050">load</span> <span m="4823490">or</span> <span m="4824630">the</span> <span m="4824760">problem</span> <span m="4825100">the</span> <span m="4825260">second</span> <span m="4825690">server</span> <span m="4826000">got</span> <span m="4826200">10%</span> <span m="4826720">too</span> <span m="4826830">much</span> <span m="4827080">load,</span> <span m="4828200">and</span> <span m="4828380">so</span> <span m="4828570">forth.</span> </p>
<p><span m="4830030">But</span> <span m="4830080">what</span> <span m="4830200">we</span> <span m="4830340">really</span> <span m="4830680">care</span> <span m="4830990">about</span> <span m="4832210">is</span> <span m="4832410">the</span> <span m="4832490">worst</span> <span m="4832980">server.</span> <span m="4835030">If</span> <span m="4835120">all</span> <span m="4835340">of</span> <span m="4835400">them</span> <span m="4835490">are</span> <span m="4835530">good</span> <span m="4835670">except</span> <span m="4835960">for</span> <span m="4836050">one,</span> <span m="4837100">you're</span> <span m="4837230">still</span> <span m="4837460">in</span> <span m="4837540">trouble.</span> <span m="4837880">Because</span> <span m="4838040">the</span> <span m="4838120">one</span> <span m="4838910">ruined</span> <span m="4839240">your</span> <span m="4839350">day.</span> <span m="4840570">Because</span> <span m="4840730">it</span> <span m="4840800">didn't</span> <span m="4840970">get</span> <span m="4841080">the</span> <span m="4841160">work</span> <span m="4841410">done.</span> <span m="4843200">So</span> <span m="4843330">what</span> <span m="4843490">do</span> <span m="4843550">you</span> <span m="4843630">do</span> <span m="4844500">to</span> <span m="4844630">bound</span> <span m="4845000">the</span> <span m="4845060">probability</span> <span m="4845720">that</span> <span m="4846160">any</span> <span m="4846450">of</span> <span m="4846530">the</span> <span m="4846600">servers</span> <span m="4847050">got</span> <span m="4847300">too</span> <span m="4847440">much</span> <span m="4847710">load,</span> <span m="4849390">any</span> <span m="4849630">of</span> <span m="4849720">the</span> <span m="4849800">10?</span> </p>
<p><span m="4852510">So</span> <span m="4853220">what</span> <span m="4853330">I</span> <span m="4853370">really</span> <span m="4853630">want</span> <span m="4853780">to</span> <span m="4853840">know</span> <span m="4854200">is</span> <span m="4855060">the</span> <span m="4855190">probability</span> <span m="4855820">that</span> <span m="4855970">the</span> <span m="4856050">worst</span> <span m="4856620">server</span> <span m="4858080">of</span> <span m="4858250">M</span> <span m="4862850">takes</span> <span m="4863100">more</span> <span m="4863300">than</span> <span m="4863440">cL</span> <span m="4863725">over</span> <span m="4864010">M.</span> <span m="4867050">Well,</span> <span m="4867460">that's</span> <span m="4867710">the</span> <span m="4867800">probability</span> <span m="4868600">that</span> <span m="4869420">the</span> <span m="4869530">first</span> <span m="4869970">one</span> <span m="4870190">has</span> <span m="4870490">more</span> <span m="4870710">than</span> <span m="4870840">cL</span> <span m="4871230">over</span> <span m="4871430">M</span> <span m="4872890">union</span> <span m="4874170">the</span> <span m="4874270">second</span> <span m="4874700">one</span> <span m="4875440">has</span> <span m="4875630">more</span> <span m="4875800">than</span> <span m="4875930">cL</span> <span m="4876320">over</span> <span m="4876510">M</span> <span m="4877480">union</span> <span m="4879030">the</span> <span m="4879200">M-th</span> <span m="4879480">one.</span> <span m="4885850">What</span> <span m="4885980">do</span> <span m="4886080">I</span> <span m="4886160">do</span> <span m="4886340">to</span> <span m="4886410">get</span> <span m="4886780">that</span> <span m="4887060">probability,</span> <span m="4889090">the</span> <span m="4889210">probability</span> <span m="4889760">of</span> <span m="4889840">a</span> <span m="4889920">union</span> <span m="4890280">of</span> <span m="4890360">events,</span> <span m="4891520">upper</span> <span m="4891760">bounded?</span> </p>
<p><span m="4893510">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4895970">PROFESSOR: Upper</span> <span m="4896240">bounded</span> <span m="4896580">by</span> <span m="4896690">the</span> <span m="4896790">sum</span> <span m="4897210">of</span> <span m="4897290">the</span> <span m="4897440">individual</span> <span m="4898010">guys.</span> <span m="4898560">It's</span> <span m="4898740">the</span> <span m="4898830">sum</span> <span m="4899820">i</span> <span m="4900250">equals</span> <span m="4900420">1</span> <span m="4900720">to</span> <span m="4900820">M</span> <span m="4902310">probability</span> <span m="4902820">Ri</span> <span m="4904290">greater than or</span> <span m="4904540">equal</span> <span m="4904740">to</span> <span m="4904870">cL</span> <span m="4905000">over</span> <span m="4905330">M.</span> <span m="4906890">And</span> <span m="4907200">so</span> <span m="4907370">that,</span> <span m="4907710">each</span> <span m="4907920">of</span> <span m="4908000">these</span> <span m="4908230">is</span> <span m="4908320">at</span> <span m="4908390">most</span> <span m="4908730">1</span> <span m="4908880">in</span> <span m="4908950">160,000.</span> <span m="4910740">This</span> <span m="4910920">is</span> <span m="4911000">at</span> <span m="4911040">most</span> <span m="4911410">M/160,000.</span> <span m="4917320">And</span> <span m="4917560">that</span> <span m="4917790">is</span> <span m="4918140">equal</span> <span m="4918480">to</span> <span m="4918620">1</span> <span m="4918765">in</span> <span m="4918910">16,000.</span> </p>
<p><span m="4923120">All</span> <span m="4923260">right,</span> <span m="4923400">so</span> <span m="4923490">now</span> <span m="4923670">we</span> <span m="4923780">have</span> <span m="4923990">the</span> <span m="4924120">answer.</span> <span m="4924440">The</span> <span m="4924550">chance</span> <span m="4924980">that</span> <span m="4925160">any</span> <span m="4925380">server</span> <span m="4925740">got</span> <span m="4926080">10%</span> <span m="4926610">load</span> <span m="4926800">or</span> <span m="4926860">more</span> <span m="4927770">is</span> <span m="4927960">1</span> <span m="4928180">in</span> <span m="4928270">16,000</span> <span m="4929320">at</span> <span m="4929480">most,</span> <span m="4931800">which</span> <span m="4931970">is</span> <span m="4932060">why</span> <span m="4933590">randomized</span> <span m="4934160">load</span> <span m="4934390">balancing</span> <span m="4934790">is</span> <span m="4934900">used</span> <span m="4935150">a</span> <span m="4935190">lot</span> <span m="4935390">in</span> <span m="4935460">practice.</span> <span m="4936700">Now</span> <span m="4936880">tomorrow,</span> <span m="4938210">you're</span> <span m="4938340">going</span> <span m="4938460">to</span> <span m="4938520">do</span> <span m="4938660">a</span> <span m="4938750">real</span> <span m="4939010">world</span> <span m="4939300">example</span> <span m="4941360">where</span> <span m="4943250">people</span> <span m="4943550">use</span> <span m="4943990">this</span> <span m="4944170">kind</span> <span m="4944380">of</span> <span m="4944450">analysis,</span> <span m="4945840">and</span> <span m="4945990">it</span> <span m="4946100">led</span> <span m="4946420">to</span> <span m="4946530">utter</span> <span m="4946810">disaster.</span> <span m="4948550">And</span> <span m="4948730">the</span> <span m="4948820">reason</span> <span m="4949200">was</span> <span m="4951380">that</span> <span m="4951630">the</span> <span m="4951700">components</span> <span m="4952380">they</span> <span m="4952500">were</span> <span m="4952760">looking</span> <span m="4953090">at</span> <span m="4953460">were</span> <span m="4953590">not</span> <span m="4953980">independent.</span> </p>
<p><span m="4955290">And</span> <span m="4955430">the</span> <span m="4955510">example</span> <span m="4955990">has</span> <span m="4956220">to</span> <span m="4956300">do</span> <span m="4956510">with</span> <span m="4956630">the</span> <span m="4956700">subprime</span> <span m="4957240">mortgage</span> <span m="4957610">disaster.</span> <span m="4957865">And</span> <span m="4958120">I</span> <span m="4958160">don't</span> <span m="4958280">have</span> <span m="4958370">time</span> <span m="4958580">today</span> <span m="4958800">to</span> <span m="4958890">go</span> <span m="4959050">through</span> <span m="4959240">it</span> <span m="4959320">all.</span> <span m="4959510">But</span> <span m="4959820">it's</span> <span m="4959960">in</span> <span m="4960030">the</span> <span m="4960100">text, and</span> <span m="4960550">you'll</span> <span m="4960580">see</span> <span m="4960750">it</span> <span m="4960850">tomorrow.</span> <span m="4962060">But</span> <span m="4962210">basically</span> <span m="4962620">what</span> <span m="4962790">happened</span> <span m="4963610">is</span> <span m="4963870">that</span> <span m="4964180">they</span> <span m="4964270">took</span> <span m="4964430">a</span> <span m="4964500">whole</span> <span m="4964720">bunch</span> <span m="4964950">of</span> <span m="4965040">loans,</span> <span m="4966580">subprime</span> <span m="4967070">loans,</span> <span m="4967950">put</span> <span m="4968140">them</span> <span m="4968260">into</span> <span m="4968380">these</span> <span m="4968560">things</span> <span m="4968770">called</span> <span m="4969050">bonds,</span> <span m="4970780">and</span> <span m="4970950">then</span> <span m="4971120">did</span> <span m="4971240">an</span> <span m="4971330">analysis</span> <span m="4971940">about</span> <span m="4972150">how</span> <span m="4972280">many</span> <span m="4972450">failures</span> <span m="4972970">they'd</span> <span m="4973100">expect</span> <span m="4973460">to</span> <span m="4973550">have.</span> <span m="4975040">And</span> <span m="4975230">they</span> <span m="4975350">assumed</span> <span m="4975760">the</span> <span m="4975820">loans</span> <span m="4976170">were</span> <span m="4976260">all</span> <span m="4976450">mutually</span> <span m="4976840">independent.</span> </p>
<p><span m="4978300">And</span> <span m="4978450">they</span> <span m="4978550">applied</span> <span m="4978920">their</span> <span m="4979040">Chernoff</span> <span m="4979530">bounds.</span> <span m="4980700">And</span> <span m="4980820">that</span> <span m="4981230">concluded</span> <span m="4981810">that</span> <span m="4982180">the</span> <span m="4982300">chances</span> <span m="4982800">of</span> <span m="4982870">being</span> <span m="4983060">off</span> <span m="4983310">from</span> <span m="4983410">the</span> <span m="4983480">expectation</span> <span m="4984180">were</span> <span m="4984300">nil,</span> <span m="4984870">like</span> <span m="4985080">e</span> <span m="4985270">to</span> <span m="4985330">the</span> <span m="4985400">minus</span> <span m="4985850">380.</span> <span m="4987890">In</span> <span m="4988040">reality,</span> <span m="4989210">the</span> <span m="4989330">loans</span> <span m="4989730">were</span> <span m="4989840">highly</span> <span m="4990250">dependent.</span> <span m="4990950">When</span> <span m="4991170">one</span> <span m="4991440">failed,</span> <span m="4991750">a</span> <span m="4991790">lot</span> <span m="4992060">tended</span> <span m="4992280">to</span> <span m="4992350">fail.</span> <span m="4993100">And</span> <span m="4993260">that</span> <span m="4993470">led</span> <span m="4993890">to</span> <span m="4994020">disaster.</span> <span m="4994570">And</span> <span m="4994660">you'll</span> <span m="4994740">go</span> <span m="4994910">through</span> <span m="4995090">some</span> <span m="4995300">of</span> <span m="4995360">the</span> <span m="4995430">math</span> <span m="4995790">on</span> <span m="4995910">that</span> <span m="4996120">tomorrow</span> <span m="4996395">in</span> <span m="4996670">recitation.</span> </p>
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